Arithmetic and Geometric Sequences and Series Practice Questions — ACT

Question 1Question →

A geometric sequence has a first term of $\frac{1}{4}$ and a common ratio of $\frac{1}{2}$ . What is the 4th term of this sequence?

\frac{1}{32}

\frac{1}{64}

\frac{1}{512}

64

\frac{2}{3}

Show answer & explanation

Answer: $\frac{1}{32}$

Answer

\frac{1}{32}

The correct answer is found by substituting

a_1 = \frac{1}{4}

,

r = \frac{1}{2}

, and

n = 4

into the geometric sequence formula

a_n = a_1 \cdot r^{n-1}

. This gives

a_4 = \frac{1}{4} \cdot (\frac{1}{2})^3 = \frac{1}{4} \cdot \frac{1}{8} = \frac{1}{32}

.

Step-by-Step Solution

1

Identify the given components of the geometric sequence.

The first term is

a_1 = \frac{1}{4}

and the common ratio is

r = \frac{1}{2}

. We want to find the 4th term, so

n = 4

.

Before performing any calculations, we must identify the values of the variables needed for the geometric sequence formula.

2

Substitute the values into the general formula for the

n

-th term of a geometric sequence,

a_n = a_1 \cdot r^{n-1}

.

a_4 = \frac{1}{4} \cdot \left(\frac{1}{2}\right)^{4-1} = \frac{1}{4} \cdot \left(\frac{1}{2}\right)^3

This formula relates the

n

-th term to the first term and the common ratio.

3

Evaluate the exponent first, and then multiply the fractions.

a_4 = \frac{1}{4} \cdot \frac{1}{8} = \frac{1}{32}

Following the order of operations, we raise the common ratio to the 3rd power before multiplying by the first term.

Key Concept

Finding the

n

-th term of a geometric sequence using the general formula

a_n = a_1 \cdot r^{n-1}

.

Question 2Question →

A sequence is defined by the formula $a_n = 2 + 3 \cdot \frac{2^n \cdot 2^{n-1}}{6}$ for all integers $n \geq 1$ . What is the value of the second term, $a_2$ ?

4

6

\frac{20}{3}

\frac{26}{7}

10

Show answer & explanation

Answer: $6$

Answer

The value of the second term is 6.

The correct answer is 6. When substituting

n = 2

into the formula, the numerator simplifies to

2^2 \cdot 2^1 = 8

. The fraction becomes

\frac{8}{6} = \frac{4}{3}

. Multiplying by 3 yields 4, and adding 2 yields 6.

Step-by-Step Solution

1

Substitute

n = 2

into the sequence formula.

a_2 = 2 + 3 \cdot \frac{2^2 \cdot 2^{2-1}}{6}

To find the second term, we replace

n

with 2 throughout the expression.

2

Simplify the exponents in the numerator.

2^2 \cdot 2^{2-1} = 4 \cdot 2^1 = 4 \cdot 2 = 8

Evaluate the exponential terms in the numerator before performing other operations.

3

Substitute the simplified numerator back and evaluate the fraction.

\frac{8}{6} = \frac{4}{3}

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, 2.

4

Multiply by 3 and add 2 to get the final answer.

2 + 3 \cdot \frac{4}{3} = 2 + 4 = 6

Perform multiplication before addition according to the order of operations.

Key Concept

Evaluating terms of a sequence given by an explicit formula involving exponent rules.

Estimated Time:1m 0s

Question 3Question →

The first term of an arithmetic sequence is $8$ , and the second term is $5$ . What is the $6$ th term of this sequence?

-15

-10

-7

-4

23

Show answer & explanation

Answer: -7

Answer

-7

The correct answer is

-7

. Since the sequence is arithmetic, it changes by a constant common difference,

d

, with each step. Calculating

d

gives

5 - 8 = -3

. To find the

6

th term, we start at the first term,

8

, and add the common difference

5

times (representing the steps from the first to the sixth term):

8 + 5(-3) = 8 - 15 = -7

.

Step-by-Step Solution

1

Determine the common difference,

d

, of the arithmetic sequence.

d = -3

Subtract the first term from the second term:

5 - 8 = -3

.

2

Set up the equation for the

n

th term of an arithmetic sequence,

a_n = a_1 + (n-1)d

.

a_6 = 8 + (6-1)(-3)

We want to find the 6th term (

n = 6

) starting with

a_1 = 8

and

d = -3

.

3

Perform the operations to find the final value.

a_6 = -7

First multiply

5 \times (-3) = -15

, then add to

8

to get

8 - 15 = -7

.

Key Concept

Finding a specific term of an arithmetic sequence using its first term and common difference.

Estimated Time:45s

Question 4Question →

The first three terms of a geometric sequence of positive numbers are $x$ , $y$ , and $z$ , in that order. An arithmetic sequence has first three terms $x$ , $y$ , and $z - 4$ , in that order. If $x = 4$ , what is the value of $y$ ?

2

4

6

8

16

Show answer & explanation

Answer: 8

Answer

8

The correct answer is 8. By defining the geometric sequence terms as

4, y, z

, we have the property

y^2 = 4z

. For the arithmetic sequence

4, y, z-4

, the common difference property gives

y - 4 = z - 4 - y

, which simplifies to

2y = z

. Substituting

z = 2y

into the first equation yields

y^2 = 8y

. Since all terms must be positive, dividing by

y

gives

y = 8

.

Step-by-Step Solution

1

Set up the equation for the geometric sequence.

y^2 = 4z

Since

4, y, z

is a geometric sequence, the ratio of consecutive terms must be equal:

\frac{y}{4} = \frac{z}{y}

, which simplifies to

y^2 = 4z

.

2

Set up the equation for the arithmetic sequence.

2y = z

Since

4, y, z - 4

is an arithmetic sequence, the difference between consecutive terms must be equal:

y - 4 = (z - 4) - y

. Simplifying this gives

2y = z

.

3

Substitute the arithmetic equation into the geometric equation.

y^2 = 8y

Substituting

z = 2y

into

y^2 = 4z

yields

y^2 = 4(2y) = 8y

.

4

Solve for the variable

y

.

y = 8

Rearranging the equation gives

y^2 - 8y = 0

, which factors as

y(y - 8) = 0

. Since the sequence consists of positive numbers,

y

must be positive, so

y = 8

.

Key Concept

Relating arithmetic and geometric sequence properties to solve a system of non-linear equations

Question 5Question →

For a certain geometric sequence, the first term is $2^3$ and the common ratio is $2^2$ . Which of the following expressions represents the 3rd term of this sequence?

2^5

2^9

2^7

2^{10}

2^{12}

Show answer & explanation

Answer: $2^7$

Answer

The 3rd term of the sequence is

2^7

.

The correct answer is

2^7

. The

n

-th term of a geometric sequence is given by

a_n = a_1 \cdot r^{n-1}

. For a first term

a_1 = 2^3

and a common ratio

r = 2^2

, the 3rd term is

a_3 = 2^3 \cdot (2^2)^{3-1} = 2^3 \cdot 2^4

. Applying the exponent addition rule for multiplying bases of the same value yields

2^{3+4} = 2^7

.

Step-by-Step Solution

1

Identify the formula for the

n

-th term of a geometric sequence.

a_n = a_1 \cdot r^{n-1}

To find any specific term in a geometric sequence, the general term formula is used.

2

Substitute the given values (

a_1 = 2^3

,

r = 2^2

, and

n = 3

) into the formula.

a_3 = 2^3 \cdot (2^2)^{3-1} = 2^3 \cdot (2^2)^2

This sets up the calculation for the 3rd term of the sequence.

3

Simplify the expression using exponent rules: first compute

(2^2)^2

, then multiply by

2^3

.

a_3 = 2^3 \cdot 2^4 = 2^{3+4} = 2^7

Power of a power rule states

(x^a)^b = x^{ab}

, and product of powers rule states

x^a \cdot x^b = x^{a+b}

.

Key Concept

Finding a specific term of a geometric sequence using the general term formula

a_n = a_1 \cdot r^{n-1}

and applying laws of exponents.

Estimated Time:45s

Question 6Question →

An infinite geometric series of positive terms has a sum of $9$ . The sum of the first two terms of the series is $8$ . What is the first term of this series?

2

3

4

6

12

Show answer & explanation

Answer: 6

Answer

The first term of the series is 6.

To find the first term of the geometric series, we set up a system of equations using the formulas for the sum of the first two terms,

S_2 = a(1+r) = 8

, and the sum of an infinite geometric series,

S_\infty = \frac{a}{1-r} = 9

. Expressing the first term as

a = 9(1-r)

and substituting this into the first equation yields

9(1-r)(1+r) = 8

, which simplifies to

9(1-r^2) = 8

. Solving for the ratio yields

r^2 = \frac{1}{9}

, which means

r = \frac{1}{3}

since all terms must be positive. Substituting

r = \frac{1}{3}

back into

a = 9(1-r)

gives the first term as

6

.

Step-by-Step Solution

1

Define variables and identify the given formulas.

Let the first term of the geometric series be

a

and the common ratio be

r

. Since the series has positive terms, we require

a > 0

and

0 < r < 1

.

This sets up the system of equations using standard geometric sequence notation.

2

Translate the given problem conditions into algebraic equations.

Equation 1 (sum of the first two terms):

S_2 = a + ar = a(1+r) = 8

.
Equation 2 (sum to infinity):

S_\infty = \frac{a}{1-r} = 9

.

We must express the two mathematical relationships given in the problem statement.

3

Express the first term

a

in terms of the common ratio

r

using Equation 2.

a = 9(1-r)

This allows for substitution into Equation 1 to solve for

r

.

4

Substitute the expression for

a

into Equation 1 and simplify.

9(1-r)(1+r) = 8 \implies 9(1-r^2) = 8 \implies 9 - 9r^2 = 8

.

Substituting simplifies the system from two variables to a single quadratic variable in terms of

r

.

5

Solve the quadratic equation for

r

.

9r^2 = 1 \implies r^2 = \frac{1}{9} \implies r = \frac{1}{3}

(since

r > 0

).

Finding the value of the common ratio is the final step before calculating the first term.

6

Substitute the value of

r

back into the expression for

a

.

a = 9\left(1 - \frac{1}{3}\right) = 9\left(\frac{2}{3}\right) = 6

.

This yields the value of the first term

a

to complete the problem.

Key Concept

Solving systems of non-linear equations using geometric sequence term and infinite sum formulas.

Alternative Method

Instead of substituting

a = 9(1-r)

into

a(1+r) = 8

, we can divide the two equations:

\frac{a(1+r)}{a/(1-r)} = \frac{8}{9} \implies (1-r)(1+r) = \frac{8}{9} \implies 1 - r^2 = \frac{8}{9}

. This directly yields

r^2 = \frac{1}{9}

without needing to isolate

a

first.

Estimated Time:2m 0s

Question 7Question →

The first term of a geometric sequence is $3$ , and the second term is $6$ . What is the $5$ th term of this sequence?

15

24

48

96

1296

Show answer & explanation

Answer: $48$

Answer

The fifth term of the sequence is

48

.

The first term of the geometric sequence is

a_1 = 3

, and the second term is

a_2 = 6

. The common ratio

r

is

6 \div 3 = 2

. The

n

th term of a geometric sequence is given by

a_n = a_1 \cdot r^{n-1}

. For the fifth term (

n=5

), the calculation is

3 \cdot 2^{5-1} = 3 \cdot 2^4 = 3 \cdot 16 = 48

. This matches the correct value.

Step-by-Step Solution

1

Find the common ratio (

r

) of the geometric sequence.

r = \frac{6}{3} = 2

The common ratio of a geometric sequence is found by dividing any term by the preceding term.

2

Apply the formula for the

n

th term of a geometric sequence,

a_n = a_1 \cdot r^{n-1}

, to find the fifth term.

a_5 = 3 \cdot 2^{5-1} = 3 \cdot 2^4

We substitute

a_1 = 3

,

r = 2

, and

n = 5

into the standard formula.

3

Evaluate the expression.

a_5 = 3 \cdot 16 = 48

Calculate

2^4 = 16

first, then multiply by

3

according to the order of operations.

Key Concept

Finding a specific term in a geometric sequence using the general formula

Question 8Question →

An entrepreneur starts a company with an operating budget of $100,000$ in its first year. For each of the next 4 years (years 2 through 5), the budget increases by a constant amount of $d$ dollars each year. For years 6 through 8, the budget increases geometrically, where the budget in year 6 is $1.5$ times the budget in year 5, and the budget increases by $50\%$ each year thereafter. If the total operating budget over the first 8 years is $1,597,500$ dollars, what is the value of $d$ ?

4,300

10,000

17,500

33,900

38,500

Show answer & explanation

Answer: $10,000$

Answer

The common difference is

10,000

.

The correct answer is found by setting up the sum of the first 5 years of the arithmetic sequence and the subsequent 3 years of the geometric sequence. Summing the expressions for all 8 years gives

1,212,500 + 38.5d

. Equating this expression to the total budget of

1,597,500

and solving for

d

yields

10,000

.

Step-by-Step Solution

1

Express the budget for the first 5 years as an arithmetic sequence and find their sum.

S_5 = 100,000 + (100,000 + d) + (100,000 + 2d) + (100,000 + 3d) + (100,000 + 4d) = 500,000 + 10d

The budget starts at

100,000

in year 1 and increases by a constant amount

d

each year through year 5.

2

Express the budgets for years 6 through 8 as a geometric sequence starting from

1.5

times the year 5 budget.

Year 6:

1.5(100,000 + 4d) = 150,000 + 6d

; Year 7:

1.5(150,000 + 6d) = 225,000 + 9d

; Year 8:

1.5(225,000 + 9d) = 337,500 + 13.5d

. The sum of these 3 years is

712,500 + 28.5d

.

The budget increases by a factor of

1.5

(or

50\%

) each year starting from year 6.

3

Combine the sums of both sequences to represent the total 8-year budget and solve for

d

.

Total =

(500,000 + 10d) + (712,500 + 28.5d) = 1,212,500 + 38.5d

. Setting this equal to the given total:

1,212,500 + 38.5d = 1,597,500 \implies 38.5d = 385,000 \implies d = 10,000

.

The total budget over the 8 years is the sum of the budgets of the individual years.

Key Concept

Combining arithmetic and geometric sequences in multi-step word problems.

Alternative Method

Instead of calculating each geometric year sequentially, the sum of the geometric sequence for years 6 to 8 can be calculated using the geometric series sum formula

S_n = a_1 \frac{1 - r^n}{1 - r}

with

a_1 = 1.5(100,000 + 4d)

and

r = 1.5

over

n = 3

terms.

Estimated Time:2m 30s

Question 9Question →

Two infinite geometric series, Series A and Series B, are defined as follows:

* Series A has a first term of $2^x$ and a common ratio of $\frac{1}{2}$ .
* Series B has a first term of $2^{x+3}$ and a common ratio of $\frac{3}{4}$ .

If the sum of Series A and Series B is 136, what is the value of $x$ ?

0.5

1

2

3

4

Show answer & explanation

Answer: $2$

Answer

2

The correct answer is the value 2. Substituting

x = 2

into the sum of Series A gives

2^{2+1} = 8

. Substituting

x = 2

into the sum of Series B gives

2^{2+5} = 128

. The total sum is

8 + 128 = 136

, which matches the given sum.

Step-by-Step Solution

1

Express the sum of Series A, denoted as

S_A

, using the infinite geometric series sum formula

S = \frac{a_1}{1-r}

.

S_A = \frac{2^x}{1 - 1/2} = \frac{2^x}{1/2} = 2 \cdot 2^x = 2^{x+1}

Since the common ratio

r = 1/2

satisfies

|r| < 1

, the infinite sum exists and can be simplified using exponent properties.

2

Express the sum of Series B, denoted as

S_B

, using the infinite geometric series sum formula.

S_B = \frac{2^{x+3}}{1 - 3/4} = \frac{2^{x+3}}{1/4} = 4 \cdot 2^{x+3} = 2^2 \cdot 2^{x+3} = 2^{x+5}

Since the common ratio

r = 3/4

satisfies

|r| < 1

, the infinite sum exists and can be simplified using exponent rules.

3

Set up the equation for the sum of both series and solve for

x

.

2^{x+1} + 2^{x+5} = 136 \implies 2 \cdot 2^x + 32 \cdot 2^x = 136 \implies 34 \cdot 2^x = 136 \implies 2^x = 4 \implies x = 2

Factoring out

2^x

from the terms allows us to isolate the exponential expression and find the value of

x

.

Key Concept

Sum of an Infinite Geometric Series and Exponential Properties

Question 10Question →

The first, third, and eleventh terms of a non-constant arithmetic sequence are the first, second, and third terms, respectively, of a geometric sequence. If the first term of the arithmetic sequence is $6$ , what is the sum of the first $4$ terms of the geometric sequence?

78

384

510

1,554

4,920

Show answer & explanation

Answer: 510

Answer

The sum of the first 4 terms of the geometric sequence is 510.

By writing the first, third, and eleventh terms of the arithmetic sequence as

6

,

6+2d

, and

6+10d

, we can set up the geometric sequence relation

(6+2d)^2 = 6(6+10d)

. Solving this quadratic equation for

d

yields

d=9

(since the sequence is non-constant,

d \neq 0

). Substituting

d=9

back gives the first three geometric terms as

6

,

24

, and

96

, which means the common ratio

r

is

4

. The sum of the first

4

terms of this geometric sequence is calculated as

6(4^4-1)/(4-1) = 510

.

Step-by-Step Solution

1

Define the terms of the arithmetic and geometric sequences.

Let the arithmetic sequence have first term

a_1 = 6

and common difference

d

. The first, third, and eleventh terms are

a_1 = 6

,

a_3 = 6 + 2d

, and

a_{11} = 6 + 10d

. These are the first three terms of the geometric sequence:

g_1 = 6

,

g_2 = 6 + 2d

, and

g_3 = 6 + 10d

.

This establishes algebraic expressions for the terms based on their positions in the sequences.

2

Set up a relation using the constant ratio of the geometric sequence and solve for

d

.

Since

g_1

,

g_2

, and

g_3

form a geometric sequence,

(g_2)^2 = g_1 \cdot g_3

. Substituting the expressions gives

(6 + 2d)^2 = 6(6 + 10d) \implies 36 + 24d + 4d^2 = 36 + 60d \implies 4d^2 - 36d = 0

. Since the sequence is non-constant (

d \neq 0

), we divide by

4d

to get

d = 9

.

Solving the equation yields the common difference of the arithmetic sequence.

3

Determine the terms and common ratio of the geometric sequence.

Using

d = 9

, the first two terms of the geometric sequence are

g_1 = 6

and

g_2 = 6 + 2(9) = 24

. The common ratio is

r = \frac{24}{6} = 4

.

Finding the common ratio allows the use of the geometric series sum formula.

4

Compute the sum of the first 4 terms of the geometric sequence.

S_4 = g_1 \frac{r^4 - 1}{r - 1} = 6 \frac{4^4 - 1}{4 - 1} = 6 \frac{256 - 1}{3} = 2(255) = 510

.

This calculates the final required sum.

Key Concept

Arithmetic and Geometric Sequences and Series

Question 11Question →

An arithmetic sequence $a_1, a_2, a_3, \dots$ has a first term $a_1 = a$ and a non-zero common difference $d$ . A geometric sequence $g_1, g_2, g_3, \dots$ has a first term $g_1 = a$ and a common ratio $r > 1$ . The third term of the arithmetic sequence is equal to the second term of the geometric sequence ( $a_3 = g_2$ ), and the seventh term of the arithmetic sequence is equal to the third term of the geometric sequence ( $a_7 = g_3$ ). If the sum of the first five terms of the arithmetic sequence is 150, what is the value of the fifth term of the geometric sequence, $g_5$ ?

240

480

800

96

810,000

Show answer & explanation

Answer: 240

Answer

The fifth term of the geometric sequence is 240.

The correct answer is 240. The term relationships

a_3 = g_2

and

a_7 = g_3

translate to

a + 2d = ar

and

a + 6d = ar^2

. Expressing the first equation as

2d = a(r - 1)

and substituting it into the second equation yields

a + 3a(r - 1) = ar^2

. Dividing by

a

gives the quadratic equation

r^2 - 3r + 2 = 0

. Since

r > 1

, we find

r = 2

, which implies

d = a/2

. The sum of the first five terms of the arithmetic sequence is

5(a + 2d) = 150

, which simplifies to

a + 2d = 30

. Substituting

d = a/2

gives

2a = 30

, so

a = 15

. The fifth term of the geometric sequence is then

g_5 = a \cdot r^4 = 15 \cdot 2^4 = 240

.

Step-by-Step Solution

1

Express the given sequence term relationships in terms of

a

,

d

, and

r

.

a_3 = a + 2d

,

a_7 = a + 6d

,

g_2 = ar

, and

g_3 = ar^2

.

To represent the equality of terms algebraically.

2

Solve the system of equations

a + 2d = ar

and

a + 6d = ar^2

for the common ratio

r

.

From the first equation,

2d = a(r - 1) \Rightarrow d = \frac{a(r - 1)}{2}

. Substituting this into the second equation gives

a + 3a(r - 1) = ar^2

. Dividing by

a

(since

a \neq 0

) yields

r^2 - 3r + 2 = 0

, which factors as

(r-1)(r-2) = 0

. Since

r > 1

, we must have

r = 2

. Thus,

d = a/2

.

To find the relationship between the arithmetic common difference, the geometric common ratio, and the first term.

3

Use the sum of the first five terms of the arithmetic sequence to solve for

a

.

S_5 = 5(a + 2d) = 150 \Rightarrow a + 2d = 30

. Substituting

d = a/2

gives

a + a = 30 \Rightarrow 2a = 30 \Rightarrow a = 15

.

To determine the numerical value of the first term.

4

Calculate the fifth term of the geometric sequence.

g_5 = a \cdot r^4 = 15 \cdot 2^4 = 15 \cdot 16 = 240

.

To find the requested term of the geometric sequence.

Key Concept

Solving systems of linear and exponential relationships using arithmetic and geometric sequence properties.

Question 12Question →

An arithmetic sequence $a_1, a_2, a_3, \dots$ has a first term $a_1$ and a common difference $d$ , where both $a_1$ and $d$ are non-zero. Let $S_n$ represent the sum of the first $n$ terms of this sequence. If the ratio $\frac{S_{3n}}{S_n}$ is equal to a constant value $C$ for all positive integers $n$ , what is the ratio of the tenth term, $a_{10}$ , to the first term, $a_1$ ?

10

21

19

9

46

Show answer & explanation

Answer: 19

Answer

The ratio of the tenth term to the first term is 19.

The sum of the first

n

terms of an arithmetic sequence is

S_n = \frac{n}{2}[2a_1 + (n-1)d]

. For

n=1

, the ratio is

\frac{S_3}{S_1} = 3 + \frac{3d}{a_1}

. For

n=2

, the ratio is

\frac{S_6}{S_2} = \frac{6a_1 + 15d}{2a_1 + d}

. Setting these equal because the ratio is constant for all positive integers

n

gives

3 + \frac{3d}{a_1} = \frac{6a_1 + 15d}{2a_1 + d}

, which simplifies to

\frac{3a_1+3d}{a_1} = \frac{6a_1+15d}{2a_1+d}

. Cross-multiplying and simplifying yields

3d^2 = 6a_1 d

. Since

d \neq 0

, we have

d = 2a_1

. The tenth term is

a_{10} = a_1 + 9d = a_1 + 9(2a_1) = 19a_1

. Thus, the ratio of the tenth term to the first term is

19

.

Step-by-Step Solution

1

Write the formula for the sum of the first

n

terms of an arithmetic sequence,

S_n = \frac{n}{2}[2a_1 + (n-1)d]

, and find the expressions for

S_1

and

S_3

.

S_1 = a_1

and

S_3 = 3a_1 + 3d

.

To evaluate the ratio

\frac{S_{3n}}{S_n}

for the case where

n = 1

.

2

Write the expressions for

S_2

and

S_6

using the arithmetic sum formula.

S_2 = 2a_1 + d

and

S_6 = 6a_1 + 15d

.

To evaluate the ratio

\frac{S_{3n}}{S_n}

for the case where

n = 2

.

3

Equate the ratio for

n = 1

to the ratio for

n = 2

since the ratio

\frac{S_{3n}}{S_n}

must be constant for all

n

.

\frac{3a_1 + 3d}{a_1} = \frac{6a_1 + 15d}{2a_1 + d}

.

To set up an algebraic equation to find the relationship between the first term

a_1

and the common difference

d

.

4

Solve the equation for

d

in terms of

a_1

by cross-multiplying and simplifying.

d = 2a_1

.

Cross-multiplying gives

(3a_1 + 3d)(2a_1 + d) = a_1(6a_1 + 15d)

, which expands to

6a_1^2 + 9a_1 d + 3d^2 = 6a_1^2 + 15a_1 d

. Subtracting

6a_1^2

and

9a_1 d

from both sides yields

3d^2 = 6a_1 d

. Since

d \neq 0

, dividing by

3d

gives

d = 2a_1

.

5

Substitute

d = 2a_1

into the formula for the tenth term,

a_{10} = a_1 + 9d

, and calculate the ratio

\frac{a_{10}}{a_1}

.

a_{10} = 19a_1

, so the ratio is

19

.

To determine the final value of the requested ratio.

Key Concept

Relating arithmetic sequence term and sum formulas through systems of algebraic equations.

Question 13Question →

An infinite geometric series has a first term of $12$ and a sum of $8$ . What is the common ratio of this series?

-\frac{1}{2}

\frac{1}{2}

\frac{1}{3}

\frac{2}{3}

\frac{5}{2}

Show answer & explanation

Answer: $-\frac{1}{2}$

Answer

The correct common ratio is

-\frac{1}{2}

.

The correct answer is

-\frac{1}{2}

. The formula for the sum of an infinite geometric series is

S = \frac{a_1}{1 - r}

. Substituting the first term

a_1 = 12

and the sum

S = 8

gives

8 = \frac{12}{1 - r}

. Multiplying both sides by

1 - r

yields

8(1 - r) = 12

, which simplifies to

8 - 8r = 12

. Subtracting

8

from both sides gives

-8r = 4

. Dividing by

-8

gives

r = -\frac{1}{2}

. Since

|-\frac{1}{2}| < 1

, the series converges.

Step-by-Step Solution

1

Write the formula for the sum of an infinite geometric series.

S = \frac{a_1}{1 - r}

This formula relates the sum (

S

), the first term (

a_1

), and the common ratio (

r

) of an infinite geometric series where

|r| < 1

.

2

Substitute the given values into the formula.

8 = \frac{12}{1 - r}

Replacing

S

with

8

and

a_1

with

12

leaves the common ratio

r

as the only unknown variable.

3

Solve for the common ratio

r

.

8(1 - r) = 12 \Rightarrow 8 - 8r = 12 \Rightarrow -8r = 4 \Rightarrow r = -\frac{4}{8} = -\frac{1}{2}

Multiplying by the denominator and isolating

r

yields the value of the common ratio.

Key Concept

Sum of an infinite geometric series

Alternative Method

Test the given choices by substituting each value of

r

back into the sum formula

S = \frac{12}{1 - r}

to see which one yields a sum of

8

. For example, testing

-\frac{1}{2}

gives

\frac{12}{1 - (-0.5)} = \frac{12}{1.5} = 8

, which matches the given sum.

Estimated Time:1m 0s

Question 14Question →

A geometric sequence consists of positive terms. The first term of the sequence is $3$ , and the sum of the first $3$ terms is $39$ . What is the value of the $4$ th term of this sequence?

33

81

120

192

243

Show answer & explanation

Answer: 81

Answer

81

The sum of the first three terms of a geometric sequence is given by

S_3 = a_1(1 + r + r^2)

. Substituting the first term

a_1 = 3

and the sum

S_3 = 39

gives

3(1 + r + r^2) = 39

, which simplifies to

r^2 + r - 12 = 0

. Factoring this quadratic equation yields

(r - 3)(r + 4) = 0

. Since the sequence consists of positive terms, the common ratio must be positive, which gives

r = 3

. The fourth term of the sequence is then calculated using the formula

a_4 = a_1 r^3 = 3 \cdot 3^3 = 81

.

Step-by-Step Solution

1

Write the expression for the sum of the first 3 terms of a geometric sequence and set it equal to the given sum.

S_3 = a_1(1 + r + r^2) = 3(1 + r + r^2) = 39

This sets up the equation needed to solve for the common ratio of the sequence.

2

Divide both sides of the equation by 3 and solve the resulting quadratic equation for the common ratio

r

.

1 + r + r^2 = 13 \implies r^2 + r - 12 = 0 \implies (r - 3)(r + 4) = 0 \implies r = 3

(since terms must be positive)

This determines the common ratio of the geometric sequence.

3

Use the formula for the

n

-th term of a geometric sequence,

a_n = a_1 r^{n-1}

, to find the value of the 4th term.

a_4 = 3 \cdot 3^3 = 3 \cdot 27 = 81

This computes the final value requested by the question.

Key Concept

Geometric sequence formulas for the sum of a finite number of terms and the n-th term

Estimated Time:1m 30s

Question 15Question →

A geometric sequence has a first term of $2^3$ and a common ratio of $2^2$ . What is the value of the 5th term of this sequence?

24

512

2,048

8,192

16,777,216

Show answer & explanation

Answer: $2,048$

Answer

The 5th term of the geometric sequence is

2,048

.

To find the 5th term of a geometric sequence, we use the formula

a_n = a_1 \cdot r^{n-1}

. Substituting the given first term

a_1 = 2^3 = 8

and the common ratio

r = 2^2 = 4

for

n = 5

gives

a_5 = 2^3 \cdot (2^2)^4

. Applying the power of a power rule,

(2^2)^4 = 2^{2 \times 4} = 2^8

. Then, multiplying the bases by adding the exponents gives

2^3 \cdot 2^8 = 2^{3+8} = 2^{11}

, which evaluates to

2,048

.

Step-by-Step Solution

1

Identify the given values and the formula for the

n

-th term of a geometric sequence.

The first term is

a_1 = 2^3 = 8

, the common ratio is

r = 2^2 = 4

, and we need to find the term for

n = 5

using the formula

a_n = a_1 \cdot r^{n-1}

.

Knowing the correct formula is necessary to compute the specific term of a geometric sequence.

2

Substitute the values into the formula to express the 5th term in terms of base 2.

a_5 = 2^3 \cdot (2^2)^{5-1} = 2^3 \cdot (2^2)^4

This substitutes the specific term number and sequence parameters into the general term formula.

3

Simplify the exponential expression and calculate the final numerical value.

a_5 = 2^3 \cdot 2^8 = 2^{3+8} = 2^{11} = 2,048

Applying exponent rules (multiplying powers of a power and adding exponents when multiplying like bases) allows us to evaluate the expression to a single number.

Key Concept

Finding a specific term in a geometric sequence using exponential properties

Estimated Time:1m 0s

Question 16Question →

A recipe calls for $\frac{1}{2}$ cup of sugar on the first day of a fermentation process. Each day after that, the ratio of the sugar added on that day to the sugar added on the previous day is $1:3$ . What is the total amount of sugar, in cups, added during the first 3 days?

\frac{13}{18}

\frac{3}{26}

\frac{2}{9}

\frac{5}{2}

\frac{21}{32}

Show answer & explanation

Answer: $\frac{13}{18}$

Answer

The total amount of sugar added is

\frac{13}{18}

cups.

The correct answer is

\frac{13}{18}

. The sugar added on Day 1 is

\frac{1}{2}

cup. Since the ratio between consecutive days is

1:3

, the common ratio is

r = \frac{1}{3}

. This gives a Day 2 amount of

\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}

cup, and a Day 3 amount of

\frac{1}{6} \times \frac{1}{3} = \frac{1}{18}

cup. Summing these three amounts using a common denominator of 18 yields

\frac{9}{18} + \frac{3}{18} + \frac{1}{18} = \frac{13}{18}

cups.

Step-by-Step Solution

1

Identify the type of sequence and the first term.

The first term is

a_1 = \frac{1}{2}

. The sequence is geometric because the ratio between the amounts added on consecutive days is constant.

The problem states that the ratio of sugar added on consecutive days is

1:3

, which establishes a common ratio for a geometric sequence.

2

Determine the common ratio and find the terms for the second and third days.

The common ratio is

r = \frac{1}{3}

. The second term is

a_2 = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}

, and the third term is

a_3 = \frac{1}{6} \times \frac{1}{3} = \frac{1}{18}

.

Each subsequent term of a geometric sequence is found by multiplying the previous term by the common ratio.

3

Calculate the sum of the first three terms.

The total sum is

S_3 = \frac{1}{2} + \frac{1}{6} + \frac{1}{18} = \frac{9}{18} + \frac{3}{18} + \frac{1}{18} = \frac{13}{18}

.

Finding the total amount requires summing the individual amounts added over the three days using a common denominator of 18.

Key Concept

Sum of a finite geometric series

Alternative Method

Instead of calculating and adding the individual terms, you can use the sum of a finite geometric series formula:

S_n = \frac{a_1(1-r^n)}{1-r}

. Substituting

a_1 = \frac{1}{2}

,

r = \frac{1}{3}

, and

n=3

yields:

S_3 = \frac{\frac{1}{2}\left(1 - \left(\frac{1}{3}\right)^3\right)}{1 - \frac{1}{3}} = \frac{\frac{1}{2}\left(1 - \frac{1}{27}\right)}{\frac{2}{3}} = \frac{\frac{1}{2}\left(\frac{26}{27}\right)}{\frac{2}{3}} = \frac{13}{27} \times \frac{3}{2} = \frac{13}{18}

.

Estimated Time:1m 30s

Question 17Question →

The first term of an arithmetic sequence is $\frac{1}{2}$ , and the third term of the sequence is $\frac{5}{6}$ . What is the sum of the first 6 terms of this sequence?

\frac{9}{4}

3

5

\frac{11}{2}

\frac{21}{2}

Show answer & explanation

Answer: $\frac{11}{2}$

Answer

The sum of the first 6 terms is

\frac{11}{2}

.

To find the sum of the first 6 terms, we first determine the common difference

d

from the given terms:

a_3 = a_1 + 2d \Rightarrow \frac{5}{6} = \frac{1}{2} + 2d \Rightarrow 2d = \frac{1}{3} \Rightarrow d = \frac{1}{6}

. Then, we apply the arithmetic series sum formula:

S_6 = \frac{6}{2}[2(\frac{1}{2}) + 5(\frac{1}{6})] = 3[1 + \frac{5}{6}] = 3(\frac{11}{6}) = \frac{11}{2}

.

Step-by-Step Solution

1

Find the common difference

d

using the formula for the

n

-th term of an arithmetic sequence

a_n = a_1 + (n - 1)d

with the given values

a_1 = \frac{1}{2}

and

a_3 = \frac{5}{6}

.

2d = \frac{5}{6} - \frac{1}{2} \Rightarrow 2d = \frac{1}{3} \Rightarrow d = \frac{1}{6}

We need to find the common difference to determine the subsequent terms and calculate the sum of the sequence.

2

Use the sum formula

S_n = \frac{n}{2}[2a_1 + (n - 1)d]

with

n = 6

,

a_1 = \frac{1}{2}

, and

d = \frac{1}{6}

to find the sum of the first 6 terms.

S_6 = \frac{6}{2}[2(\frac{1}{2}) + (6 - 1)(\frac{1}{6})] = 3[1 + \frac{5}{6}] = 3(\frac{11}{6}) = \frac{11}{2}

This formula sums the first 6 terms of the arithmetic sequence directly.

Key Concept

Sum of the first

n

terms of an arithmetic sequence using fractional terms

Alternative Method

Alternatively, you can list the first 6 terms of the sequence and add them directly:

\frac{3}{6}, \frac{4}{6}, \frac{5}{6}, \frac{6}{6}, \frac{7}{6}, \frac{8}{6}

. Adding these gives

\frac{33}{6} = \frac{11}{2}

.

Estimated Time:1m 30s

Question 18Question →

On the first day of a research project, a student analyzes $\frac{1}{4}$ of a dataset. On each subsequent day, the student analyzes a fraction of the dataset that is exactly $\frac{2}{3}$ of the fraction analyzed on the previous day. What fraction of the dataset does the student analyze on the 4th day of the project?

\frac{1}{216}

\frac{4}{81}

\frac{2}{27}

\frac{1}{2}

\frac{9}{31}

Show answer & explanation

Answer: $\frac{2}{27}$

Answer

The fraction of the dataset analyzed on the 4th day is

\frac{2}{27}

.

The problem describes a geometric sequence where the first term is the fraction of the dataset analyzed on the first day, which is

\frac{1}{4}

, and the common ratio is the multiplier for each subsequent day, which is

\frac{2}{3}

. The general formula for the

n

-th term of a geometric sequence is

a_n = a_1 \cdot r^{n-1}

. For the 4th day, we evaluate

a_4 = \frac{1}{4} \cdot \left(\frac{2}{3}\right)^{4-1} = \frac{1}{4} \cdot \left(\frac{2}{3}\right)^3 = \frac{1}{4} \cdot \frac{8}{27} = \frac{2}{27}

.

Step-by-Step Solution

1

Identify the type of sequence and its parameters from the word problem.

This is a geometric sequence with the first term

a_1 = \frac{1}{4}

and a common ratio

r = \frac{2}{3}

.

Each day's fraction is a constant multiple of the previous day's fraction, which characterizes a geometric sequence.

2

Set up the formula for the

n

-th term of a geometric sequence to find the term for the 4th day.

a_n = a_1 \cdot r^{n-1} \implies a_4 = \frac{1}{4} \cdot \left(\frac{2}{3}\right)^{4-1}

To find the fraction analyzed on the 4th day, we evaluate the 4th term of the sequence (

n = 4

).

3

Simplify the exponent and calculate the final fraction.

a_4 = \frac{1}{4} \cdot \left(\frac{2}{3}\right)^3 = \frac{1}{4} \cdot \frac{8}{27} = \frac{2}{27}

Performing exponentiation before multiplication satisfies the order of operations and yields the correct fraction.

Key Concept

Modeling real-world scenarios using the general term formula of geometric sequences.

Question 19Question →

A geometric sequence has a second term of $-\frac{3}{2}$ and a fifth term of $12$ . What is the eighth term of this sequence?

-96

-48

\frac{51}{2}

96

192

Show answer & explanation

Answer: $-96$

Answer

-96

The correct term is

-96

. First, find the common ratio

r

by taking the ratio of the fifth term to the second term:

\frac{a_5}{a_2} = \frac{a_1 r^4}{a_1 r} = r^3

. Substituting the given values,

r^3 = \frac{12}{-3/2} = -8

, which gives

r = -2

. To find the eighth term, multiply the fifth term by the common ratio cubed:

a_8 = a_5 r^3 = 12 \times (-2)^3 = 12 \times (-8) = -96

.

Step-by-Step Solution

1

Set up the ratio between the fifth term and the second term using the geometric sequence formula

a_n = a_1 r^{n-1}

.

\frac{a_5}{a_2} = \frac{a_1 r^4}{a_1 r} = r^3

This allows us to isolate the common ratio

r

without needing to calculate the first term

a_1

first.

2

Substitute the given values into the ratio and solve for

r

.

r^3 = \frac{12}{-3/2} = 12 \times \left(-\frac{2}{3}\right) = -8 \Rightarrow r = -2

Finding the common ratio is necessary to compute any subsequent terms in the sequence.

3

Use the common ratio to find the eighth term

a_8

by multiplying the fifth term

a_5

by

r^3

.

a_8 = a_5 r^{8-5} = 12 \times (-2)^3 = 12 \times (-8) = -96

Since

a_8 = a_5 r^3

, multiplying

12

by

-8

directly gives the eighth term.

Key Concept

Finding terms in a geometric sequence using the common ratio.

Question 20Question →

An arithmetic sequence has a first term of $2$ and a common difference of $d$ . A geometric sequence has a first term of $4$ and a common ratio of $r$ . The third term of the arithmetic sequence is equal to the third term of the geometric sequence. If $d = r$ and $d \neq 1$ , what is the value of $d$ ?

2

\frac{1}{2}

-1

-\frac{1}{2}

\frac{1}{3}

Show answer & explanation

Answer: $-\frac{1}{2}$

Answer

-\frac{1}{2}

The correct answer is

-\frac{1}{2}

. The third term of the arithmetic sequence is

a_3 = a_1 + 2d = 2 + 2d

. The third term of the geometric sequence is

g_3 = g_1 r^2 = 4r^2

. Given that

d = r

, we set the two terms equal:

2 + 2d = 4d^2

. Rearranging and dividing by

2

yields

2d^2 - d - 1 = 0

, which factors as

(2d + 1)(d - 1) = 0

. Since the problem specifies that

d \neq 1

, the only valid solution is

d = -\frac{1}{2}

.

Step-by-Step Solution

1

Write the formulas for the third term of both the arithmetic and geometric sequences.

For the arithmetic sequence:

a_3 = a_1 + (3 - 1)d = 2 + 2d

. For the geometric sequence:

g_3 = g_1 \cdot r^{3 - 1} = 4r^2

.

This establishes the algebraic expressions for the third terms using the given first terms.

2

Substitute

d

for

r

and set the two expressions equal to each other.

Since

d = r

, we substitute

d

into the geometric term to get

g_3 = 4d^2

. Setting them equal gives

2 + 2d = 4d^2

.

The problem states that the third terms are equal and that the common difference equals the common ratio.

3

Solve the quadratic equation

4d^2 - 2d - 2 = 0

for

d

.

Divide the equation by

2

to get

2d^2 - d - 1 = 0

. Factoring this gives

(2d + 1)(d - 1) = 0

. The roots are

d = -\frac{1}{2}

and

d = 1

. Since the problem specifies

d \neq 1

, we have

d = -\frac{1}{2}

.

Solving the quadratic equation gives the possible values of the common difference, and the constraint rules out

d = 1

.

Key Concept

Relating arithmetic and geometric sequence terms and solving the resulting quadratic equation.

Alternative Method

Instead of factoring, the quadratic formula can be used to solve

2d^2 - d - 1 = 0

:

d = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-1)}}{2(2)} = \frac{1 \pm \sqrt{9}}{4} = \frac{1 \pm 3}{4}

. This yields

d = 1

and

d = -\frac{1}{2}

. Since the problem specifies

d \neq 1

, the only valid solution is

d = -\frac{1}{2}

.

Estimated Time:1m 30s

Arithmetic and Geometric Sequences and Series

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution