Question

What is the 9th term of a geometric sequence for which a1=5
and a2=25?

a9=

Answer 1

In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio, denoted as \( r \).

Given:
- \( a_1 = 5 \)
- \( a_2 = 25 \)

The common ratio \( r \) can be found using the formula for the second term:
\[
a_2 = a_1 \cdot r
\]
Plugging in the given values:
\[
25 = 5 \cdot r
\]
To solve for \( r \):
\[
r = \frac{25}{5} = 5
\]

Now that we have \( r \), we can find the \( n \)-th term of the geometric sequence using the formula:
\[
a_n = a_1 \cdot r^{n-1}
\]

To find the 9th term (\( a_9 \)):
\[
a_9 = a_1 \cdot r^{9-1} = 5 \cdot 5^{8}
\]

Calculating \( 5^{8} \):
\[
5^{8} = 390625
\]

Now, finding \( a_9 \):
\[
a_9 = 5 \cdot 390625 = 1953125
\]

Thus, the 9th term \( a_9 \) is:
\[
\boxed{1953125}
\]