For a certain geometric sequence, the first term is 2^3 and the common ratio is 2^2. Which… — ACT

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?

A
$2^5$
B
$2^9$
$2^7$ Answer
D
$2^{10}$
E
$2^{12}$

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

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.

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.

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

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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