  Find the 

8

th term of the geometric sequence whose common ratio is 

13

 and whose first term is 

5

.

To find the \( 8 \)-th term of a geometric sequence, we use the formula for the \( n \)-th term of a geometric sequence:

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

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

Given:
- \( a_1 = 5 \) (the first term),
- \( r = 13 \) (the common ratio),
- \( n = 8 \) (the term number you want to find).

Now substitute the values into the formula:

\[
a_8 = 5 \cdot 13^{8-1} = 5 \cdot 13^7
\]

Now, calculate \( 13^7 \):

\[
13^7 = 62748517
\]

Thus:

\[
a_8 = 5 \cdot 62748517 = 313742585
\]

Therefore, the 8th term of the geometric sequence is \( 313742585 \).

Do you want more details or have any questions? Here are some related questions:

1. What is the formula for the sum of the first \( n \) terms of a geometric sequence?
2. How does the formula for the \( n \)-th term change if the sequence is decreasing?
3. Can you explain how the common ratio affects the growth of the sequence?
4. What is the difference between an arithmetic and a geometric sequence?
5. How would you find the geometric mean between two numbers?

**Tip**: In geometric sequences, the common ratio dictates how quickly the terms grow or shrink—larger ratios lead to rapid increases!

Find the 8th term of the geometric sequence whose common ratio is 13 and whose first term is 5.

Discover how to find the 8th term of a geometric sequence where the first term is 5 and the common ratio is 13. This step-by-step guide covers key algebraic concepts, including the formula for the nth term of a geometric sequence. Perfect for students in Grades 9-11.

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