A geometric sequence has a first term of 2^3 and a common ratio of 2^2. What is the value … — ACT

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?

A
$24$
B
$512$
$2,048$ Answer
D
$8,192$
E
$16,777,216$

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

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.

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.

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

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