Question

In a geometric sequence, the first term, a, start subscript, 1, end subscript, commaa
1
​
, is equal to 8, comma8, and the fifth term, a, start subscript, 5, end subscript, commaa
5
​
, is equal to 128, .128. Which number represents the common ratio of the geometric sequence?
Answer
Multiple Choice Answers
r, equals, 1r=1
r, equals, 2r=2
r, equals, 3r=3
r, equals, 4r=4

Answer 1

In a geometric sequence, the \( n \)-th term is given by the formula:

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

where:
- \( a_n \) is the \( n \)-th term,
- \( a_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.

Given:
- The first term \( a_1 = 8 \)
- The fifth term \( a_5 = 128 \)

Using the formula for \( a_5 \):

\[
a_5 = a_1 \cdot r^{5-1} = 8 \cdot r^4
\]

We can equate this to the known value of \( a_5 \):

\[
8 \cdot r^4 = 128
\]

To solve for \( r^4 \), we divide both sides by 8:

\[
r^4 = \frac{128}{8} = 16
\]

Next, we take the fourth root of both sides to find \( r \):

\[
r = \sqrt[4]{16}
\]

Since \( 16 = 2^4 \), we have:

\[
r = 2
\]

Thus, the common ratio \( r \) is:

\[
\boxed{2}
\]

This corresponds to the option \( r = 2 \).