Arithmetic and Geometric Sequences and Series Alıştırma Soruları — GMAT

Soru 1Soru →

An arithmetic sequence $a_1, a_2, a_3, \dots$ with a positive common difference $d$ and a geometric sequence $b_1, b_2, b_3, \dots$ with a positive common ratio $r$ both have the same positive first term ( $a_1 = b_1 > 0$ ). If the 3rd term of the arithmetic sequence equals the 3rd term of the geometric sequence ( $a_3 = b_3$ ), and the 7th term of the arithmetic sequence equals the 5th term of the geometric sequence ( $a_7 = b_5$ ), what is the value of $r$ ?

\sqrt{2}

\sqrt{3}

2

\frac{2\sqrt{3}}{3}

2\sqrt{2}

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Cevap: $\sqrt{2}$

Cevap

\sqrt{2}

Using the formulas for the

n

-th terms,

a_3 = a_1 + 2d

and

b_3 = a_1 r^2

, giving

2d = a_1(r^2 - 1)

. Substituting

6d = 3a_1(r^2 - 1)

into

a_7 = a_1 + 6d = a_1 r^4

yields

a_1 + 3a_1(r^2 - 1) = a_1 r^4

. Dividing by

a_1 > 0

gives

1 + 3r^2 - 3 = r^4

, which simplifies to

r^4 - 3r^2 + 2 = 0

. Factoring gives

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

. Since

d > 0

and

a_1 > 0

, we must have

r^2 > 1

, eliminating

r^2 = 1

. Therefore,

r^2 = 2

, and since

r > 0

,

r = \sqrt{2}

.

Adım Adım Çözüm

1

Express sequence terms in terms of first term

a_1

, common difference

d

, and common ratio

r

a_3 = a_1 + 2d

,

a_7 = a_1 + 6d

,

b_3 = a_1 r^2

, and

b_5 = a_1 r^4

Apply the standard formulas for the

n

-th term of arithmetic (

a_n = a_1 + (n-1)d

) and geometric (

b_n = b_1 r^{n-1}

) sequences.

2

Set up equations based on given equality of terms

Equation 1:

a_1 + 2d = a_1 r^2 \implies 2d = a_1(r^2 - 1)

;
Equation 2:

a_1 + 6d = a_1 r^4

Translate the given conditions

a_3 = b_3

and

a_7 = b_5

into algebraic relations.

3

Substitute

2d

from Equation 1 into Equation 2

a_1 + 3(2d) = a_1 + 3a_1(r^2 - 1) = a_1 r^4

Express

6d

as

3(2d)

to eliminate

d

from the system.

4

Divide by

a_1

(since

a_1 > 0

) and simplify to solve for

r

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

Reduce the equation to a quadratic in terms of

r^2

.

5

Determine the valid root for

r

r^2 = 2 \implies r = \sqrt{2}

(since

r > 0

and

d > 0

implies

r^2 > 1

)

Since

d > 0

and

a_1 > 0

,

2d = a_1(r^2 - 1) > 0

, which requires

r^2 > 1

. Thus

r^2 = 1

is rejected, leaving

r^2 = 2

.

Anahtar Kavram

Arithmetic and Geometric Sequences Alignment

Tahmini Süre:2m 0s

Soru 2Soru →

Four numerical quantities $K$ , $L$ , $M$ , and $N$ are defined based on arithmetic and geometric sequences as follows. Arrange these four quantities in ascending order (from smallest to largest value):

- **Quantity $K$ **: The sum of the first 5 terms of an arithmetic sequence with first term $a_1 = 2$ and common difference $d = 3$ .
- **Quantity $L$ **: The 4th term of a geometric sequence with first term $b_1 = 3$ and common ratio $r = 2$ .
- **Quantity $M$ **: The sum of an infinite geometric series with first term $c_1 = 18$ and common ratio $r = \frac{1}{2}$ .
- **Quantity $N$ **: The 7th term of an arithmetic sequence with first term $d_1 = 50$ and common difference $d = -4$ .

Which of the following represents the correct ascending order of the four quantities?

Öğeleri doğru sıraya koymak için sürükleyin

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Cevap

The correct ascending order of the quantities is Quantity L (24) < Quantity N (26) < Quantity M (36) < Quantity K (40).

Evaluating each quantity gives Quantity L = 24, Quantity N = 26, Quantity M = 36, and Quantity K = 40. Arranging these in ascending numerical order produces the sequence Quantity L, Quantity N, Quantity M, Quantity K.

Adım Adım Çözüm

1

Calculate Quantity K

K = 40

The sum of an arithmetic sequence is given by

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

. For

n = 5

,

a_1 = 2

, and

d = 3

,

S_5 = \frac{5}{2}[2(2) + (5-1)(3)] = \frac{5}{2}[4 + 12] = \frac{5}{2}(16) = 40

.

2

Calculate Quantity L

L = 24

The

n

-th term of a geometric sequence is

b_n = b_1 \cdot r^{n-1}

. For

n = 4

,

b_1 = 3

, and

r = 2

,

b_4 = 3 \cdot 2^{4-1} = 3 \cdot 2^3 = 3 \cdot 8 = 24

.

3

Calculate Quantity M

M = 36

The sum of an infinite geometric series with

|r| < 1

is

S_\infty = \frac{c_1}{1 - r}

. For

c_1 = 18

and

r = \frac{1}{2}

,

S_\infty = \frac{18}{1 - 1/2} = \frac{18}{1/2} = 36

.

4

Calculate Quantity N

N = 26

The

n

-th term of an arithmetic sequence is

d_n = d_1 + (n-1)d

. For

n = 7

,

d_1 = 50

, and

d = -4

,

d_7 = 50 + (7-1)(-4) = 50 + 6(-4) = 50 - 24 = 26

.

5

Compare the evaluated quantities to arrange them in ascending order

24 < 26 < 36 < 40

, which corresponds to

L < N < M < K

Comparing the numeric values directly yields

24 \text{ (L)} < 26 \text{ (N)} < 36 \text{ (M)} < 40 \text{ (K)}

.

Anahtar Kavram

Arithmetic and Geometric Sequence Formulas

Soru 3Soru →

The sum of the first three terms of an increasing geometric sequence of positive numbers is $21$ , and the sum of the squares of these same three terms is $189$ . What is the first term of the sequence?

2

3

6

9

12

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The first term of the sequence is 3.

By setting up the equations

a(1+r+r^2) = 21

and

a^2(1+r^2+r^4) = 189

, we use the identity

1+r^2+r^4 = (1+r+r^2)(1-r+r^2)

to find

a(1-r+r^2) = 9

. Subtracting this from the first equation gives

2ar = 12

, so

ar = 6

. Solving for

r

gives

r = 2

for an increasing sequence, which yields

a = 3

.

Adım Adım Çözüm

1

Set up the algebraic equations for the sum of terms and sum of squared terms.

Let the first three terms be

a

,

ar

, and

ar^2

. The given conditions yield

a(1 + r + r^2) = 21

and

a^2(1 + r^2 + r^4) = 189

.

Standard representation of a geometric sequence with first term

a

and common ratio

r

.

2

Factor the sum of squares expression using algebraic identities.

Note that

1 + r^2 + r^4 = (1 + r + r^2)(1 - r + r^2)

. Thus,

a^2(1 + r + r^2)(1 - r + r^2) = 189

.

Factoring allows substitution of the first equation into the second.

3

Substitute

a(1 + r + r^2) = 21

into the factored equation.

Substituting

21

gives

21 \cdot a(1 - r + r^2) = 189 \implies a(1 - r + r^2) = 9

.

Simplifies the second-degree term expression to a system of two linear equations in terms of

a

and

ar

.

4

Subtract the simplified equation from the initial sum equation to isolate

ar

.

a(1 + r + r^2) - a(1 - r + r^2) = 21 - 9 \implies 2ar = 12 \implies ar = 6

.

Eliminating terms isolates the product of the first term and ratio, which is the second term.

5

Solve for

r

and find

a

.

Substitute

a = \frac{6}{r}

into

a(1 + r + r^2) = 21

to get

6r^2 - 15r + 6 = 0

, which factors as

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

. Since the sequence is increasing,

r = 2

, yielding

a = 3

.

Determines the specific parameters satisfying the increasing constraint.

Anahtar Kavram

Properties and algebraic manipulation of terms in geometric sequences and series.

Tahmini Süre:2m 0s

Soru 4Soru →

Sequence $A$ is an arithmetic sequence with first term $a_1 = 5$ and common difference $d = 3$ .
Sequence $B$ is a geometric sequence with first term $b_1 = 2$ and common ratio $r = \sqrt{3}$ .

Arrange the four quantities defined below in ascending order (from smallest to largest value).

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct ascending order is Quantity K, followed by Quantity N, Quantity M, and finally Quantity L.

Evaluating each expression gives Quantity K = 47, Quantity N ≈ 142.07, Quantity M = 162, and Quantity L = 185. Comparing these values from smallest to largest yields the order K, N, M, L.

Adım Adım Çözüm

1

Calculate Quantity K (

a_{15}

for Sequence A)

a_{15} = 5 + (15 - 1) \times 3 = 47

The

n

-th term of an arithmetic sequence is given by

a_n = a_1 + (n - 1)d

.

2

Calculate Quantity L (

S_{10}

for Sequence A)

S_{10} = \frac{10}{2} \times [2(5) + (10 - 1) \times 3] = 5 \times (10 + 27) = 185

The sum of the first

n

terms of an arithmetic sequence is given by

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

.

3

Calculate Quantity M (

b_9

for Sequence B)

b_9 = 2 \times (\sqrt{3})^{9 - 1} = 2 \times (\sqrt{3})^8 = 2 \times 3^4 = 162

The

n

-th term of a geometric sequence is given by

b_n = b_1 r^{n-1}

.

4

Calculate and approximate Quantity N (

S_6

for Sequence B)

S_6 = \frac{2((\sqrt{3})^6 - 1)}{\sqrt{3} - 1} = \frac{2(27 - 1)}{\sqrt{3} - 1} = \frac{52}{\sqrt{3} - 1} = 52(\sqrt{3} + 1) \approx 52 \times 2.732 = 142.07

The sum of a geometric series is

S_n = \frac{b_1(r^n - 1)}{r - 1}

. Rationalizing the denominator yields

52(\sqrt{3} + 1)

.

5

Compare all four values to establish ascending order

47 < 142.07 < 162 < 185

, which corresponds to

K < N < M < L

Ordering the numerical outputs from least to greatest gives the required sequence.

Anahtar Kavram

Calculating specific terms and sums of arithmetic and geometric sequences using explicit formulas and ordering calculated quantities.

Soru 5Soru →

An arithmetic sequence and a geometric sequence both have a first term equal to $2$ . The 3rd term of the arithmetic sequence is equal to the 2nd term of the geometric sequence, and the 7th term of the arithmetic sequence is equal to the 3rd term of the geometric sequence. If the common ratio of the geometric sequence is not equal to $1$ , what is the sum of the first $10$ terms of the arithmetic sequence?

20

30

54

65

130

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Cevap: 65

Cevap

The sum of the first 10 terms of the arithmetic sequence is 65.

By writing the equations for the given sequence terms (

2 + 2d = 2r

and

2 + 6d = 2r^2

), we substitute

d = r - 1

into the quadratic equation to find

r^2 - 3r + 2 = 0

. Since

r \neq 1

, we find

r = 2

, which gives a common difference

d = 1

. Applying the sum formula for an arithmetic sequence for

n = 10

yields

S_{10} = \frac{10}{2}(2(2) + 9(1)) = 5(13) = 65

.

Adım Adım Çözüm

1

Express the 3rd and 7th terms of the arithmetic sequence and 2nd and 3rd terms of the geometric sequence.

Arithmetic terms:

a_3 = 2 + 2d

and

a_7 = 2 + 6d

. Geometric terms:

b_2 = 2r

and

b_3 = 2r^2

.

The

n

-th term of an arithmetic sequence is

a_n = a_1 + (n-1)d

and of a geometric sequence is

b_n = b_1 r^{n-1}

.

2

Set up the system of equations given by the problem.

2 + 2d = 2r \implies 1 + d = r \implies d = r - 1

, and

2 + 6d = 2r^2 \implies 1 + 3d = r^2

.

We are given

a_3 = b_2

and

a_7 = b_3

with

a_1 = b_1 = 2

.

3

Substitute

d = r - 1

into the second equation and solve for

r

.

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

. Since

r \neq 1

,

r = 2

.

Solving the quadratic equation yields two roots, but the problem excludes

r = 1

.

4

Determine the common difference

d

and compute the sum of the first 10 terms of the arithmetic sequence.

d = 2 - 1 = 1

.

S_{10} = \frac{10}{2}[2(2) + (10 - 1)(1)] = 5(4 + 9) = 65

.

The sum of the first

n

terms of an arithmetic sequence is given by

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

.

Anahtar Kavram

Combining Arithmetic and Geometric Sequence Formulas

Tahmini Süre:2m 0s

Soru 6Soru →

Consider the four values defined below based on arithmetic and geometric sequences. Arrange these four items in ascending order (from smallest numerical value to largest numerical value).

Öğeleri doğru sıraya koymak için sürükleyin

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Cevap

The correct ascending order from smallest to largest value is: (1) The 7th term of the geometric sequence (

192

), (2) The sum of the first 8 terms of the arithmetic sequence (

208

), (3) The sum of the infinite geometric series (

225

), and (4) The 50th term of the arithmetic sequence (

250

).

Evaluating each item yields: the 7th term of the geometric sequence equals 192, the sum of the first 8 terms of the arithmetic sequence equals 208, the sum of the infinite geometric series equals 225, and the 50th term of the arithmetic sequence equals 250. Placing these in ascending numerical order produces 192 < 208 < 225 < 250.

Adım Adım Çözüm

1

Calculate the value of the 7th term of the geometric sequence

Using

g_7 = 3 \cdot 2^{7-1} = 3 \cdot 64 = 192

.

The

n\text{th}

term of a geometric sequence is given by

g_n = g_1 r^{n-1}

.

2

Calculate the sum of the first 8 terms of the arithmetic sequence

Using

S_8 = \frac{8}{2}[2(5) + (8-1)6] = 4[10 + 42] = 208

.

The sum of the first

n

terms of an arithmetic sequence is given by

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

.

3

Calculate the sum of the infinite geometric series

Using

S_\infty = \frac{150}{1 - 1/3} = \frac{150}{2/3} = 225

.

The sum of an infinite geometric series with

|r| < 1

is

S_\infty = \frac{a_1}{1-r}

.

4

Determine the common difference and the 50th term of the arithmetic sequence

Find

d = \frac{35 - 15}{7 - 3} = 5

, then

a_{50} = 15 + (50 - 3)5 = 15 + 235 = 250

.

Linear spacing between terms

a_k

and

a_m

yields

a_m - a_k = (m - k)d

.

5

Compare the calculated values to order them from smallest to largest

192 < 208 < 225 < 250

.

Arranging the quantities according to their numerical values gives the final sorted sequence.

Anahtar Kavram

Arithmetic and Geometric Sequences and Series

Soru 7Soru →

In a geometric sequence of positive numbers, the first term is $3$ and the common ratio is $2$ . What is the value of the 5th term of this sequence?

48

96

1,296

24

16

Cevabı ve açıklamayı göster

Cevap: $48$

Cevap

The 5th term of the sequence is

48

.

The

n

-th term of a geometric sequence is given by

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

. Substituting

a_1 = 3

,

r = 2

, and

n = 5

gives

a_5 = 3 \cdot 2^4 = 3 \cdot 16 = 48

.

Adım Adım Çözüm

1

Identify the formula for the

n

-th term of a geometric sequence.

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

In any geometric sequence, each term is obtained by multiplying the previous term by the common ratio

r

.

2

Substitute the given values into the formula.

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

The first term

a_1 = 3

, common ratio

r = 2

, and term number

n = 5

.

3

Calculate the exponent and final value.

a_5 = 3 \cdot 16 = 48

Evaluate

2^4 = 16

first according to order of operations, then multiply by 3.

Anahtar Kavram

Geometric Sequence

n

-th Term Formula

Soru 8Soru →

Four sequence-derived values $S_1, S_2, S_3,$ and $S_4$ are defined below. Arrange these values in order from smallest to largest.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order from smallest to largest is

S_1 < S_3 < S_4 < S_2

(corresponding to

55 < 64 < 72 < 80

).

Computing each value yields

S_1 = 55

,

S_2 = 80

,

S_3 = 64

, and

S_4 = 72

. Arranging these values in ascending order results in

55 < 64 < 72 < 80

, corresponding to

S_1 < S_3 < S_4 < S_2

.

Adım Adım Çözüm

1

Calculate the value of

S_1

S_1 = 55

Using the sum formula for an arithmetic sequence

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

, we have

S_5 = \frac{5}{2}[2(3) + (5-1)4] = \frac{5}{2}[6 + 16] = 55

.

2

Calculate the value of

S_2

S_2 = 80

Using the sum formula for a finite geometric sequence

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

, we have

S_4 = \frac{2(3^4 - 1)}{3 - 1} = \frac{2(80)}{2} = 80

.

3

Calculate the value of

S_3

S_3 = 64

Using the formula for the

n

-th term of an arithmetic sequence

a_n = a_1 + (n-1)d

, we have

c_{10} = 10 + (10-1)6 = 10 + 54 = 64

.

4

Calculate the value of

S_4

S_4 = 72

Using the sum formula for an infinite geometric series

S_\infty = \frac{a_1}{1 - r}

, we have

S_4 = \frac{48}{1 - 1/3} = \frac{48}{2/3} = 72

.

5

Order the computed values from smallest to largest

S_1 (55) < S_3 (64) < S_4 (72) < S_2 (80)

Comparing the numerical values gives

55 < 64 < 72 < 80

.

Anahtar Kavram

Evaluation and comparison of finite and infinite arithmetic and geometric sequence terms and sums

Soru 9Soru →

An arithmetic sequence $a_1, a_2, a_3, \dots$ has a first term $a_1 = 4$ and a common difference $d = 3$ . A geometric sequence $b_1, b_2, b_3, \dots$ has a first term $b_1 = 2$ and a common ratio $r = 2$ . If the $k$ -th term of the arithmetic sequence and the $m$ -th term of the geometric sequence are both equal to $64$ , what is the value of $k + m$ ?

Cevabı ve açıklamayı göster

Cevap: 27

Cevap

The value of

k + m

is 27.

For the arithmetic sequence, the

k

-th term is

a_k = a_1 + (k-1)d

. Setting

4 + 3(k-1) = 64

gives

3(k-1) = 60

, so

k-1 = 20

and

k = 21

. For the geometric sequence, the

m

-th term is

b_m = b_1 r^{m-1}

. Setting

2 \cdot 2^{m-1} = 64

gives

2^m = 64

, which implies

m = 6

. Adding the two values gives

k + m = 21 + 6 = 27

.

Adım Adım Çözüm

1

Determine the term position

k

in the arithmetic sequence.

k = 21

Using

a_k = a_1 + (k-1)d

, set

4 + 3(k-1) = 64 \implies 3(k-1) = 60 \implies k - 1 = 20 \implies k = 21

.

2

Determine the term position

m

in the geometric sequence.

m = 6

Using

b_m = b_1 \cdot r^{m-1}

, set

2 \cdot 2^{m-1} = 64 \implies 2^m = 64 \implies m = 6

.

3

Compute the sum of the two position indices

k

and

m

.

27

k + m = 21 + 6 = 27

.

Anahtar Kavram

Calculating term indices in arithmetic and geometric sequences using general term formulas

Tahmini Süre:1m 30s

Soru 10Soru →

In an arithmetic sequence, the sum of the first $4$ terms is $28$ and the sum of the first $8$ terms is $88$ . What is the $10\text{th}$ term of this sequence?

18

20

22

24

26

Cevabı ve açıklamayı göster

Cevap: $22$

Cevap

The 10th term of the sequence is 22.

Using the arithmetic series sum formula

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

, the conditions yield two linear equations:

2a_1 + 3d = 14

and

2a_1 + 7d = 22

. Subtracting these equations gives

4d = 8

, so

d = 2

. Substituting

d = 2

back into

2a_1 + 3(2) = 14

yields

a_1 = 4

. The 10th term is then calculated as

a_1 + 9d = 4 + 9(2) = 22

.

Adım Adım Çözüm

1

Express the given sums using the arithmetic series sum formula

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

.

For

S_4 = 28

:

\frac{4}{2}(2a_1 + 3d) = 28 \implies 2a_1 + 3d = 14

.
For

S_8 = 88

:

\frac{8}{2}(2a_1 + 7d) = 88 \implies 2a_1 + 7d = 22

.

Setting up linear equations in terms of the first term

a_1

and common difference

d

allows us to solve for both sequence parameters.

2

Subtract the first equation from the second equation to solve for

d

.

(2a_1 + 7d) - (2a_1 + 3d) = 22 - 14 \implies 4d = 8 \implies d = 2

.

Eliminating

a_1

isolates the common difference

d

.

3

Substitute

d = 2

back into the first equation to solve for

a_1

.

2a_1 + 3(2) = 14 \implies 2a_1 + 6 = 14 \implies 2a_1 = 8 \implies a_1 = 4

.

Finding

a_1

completes the essential parameters of the sequence.

4

Calculate the 10th term using the formula

a_n = a_1 + (n-1)d

.

a_{10} = 4 + (10 - 1)(2) = 4 + 18 = 22

.

Evaluating the formula at

n = 10

provides the target term.

Anahtar Kavram

Arithmetic sequence term and series sum formulas.

Tahmini Süre:2m 0s

Arithmetic and Geometric Sequences and Series

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