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!