To solve this problem, we will 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,
• $n$ is the term number.

Given:

• $a_8 = 65,536$,
• $r = 4$,
• $n = 8$.

We need to calculate $a_1$. Substituting into the formula:

$65,536 = a_1 \cdot 4^{7}$

Now, let's solve for $a_1$.The first term $a_1$ of the geometric sequence is $4$.

Would you like more details or have any other questions?

Here are 5 related questions for further exploration:

1. How do you calculate the sum of the first 8 terms of a geometric sequence?
2. How does the formula change for an arithmetic sequence?
3. What happens to the sequence if the common ratio is less than 1?
4. How can you calculate the nth term of a geometric series if you know the sum and the first term?
5. How do geometric sequences apply in real-world scenarios, like finance or physics?

Tip: In a geometric sequence, if the common ratio is greater than 1, the terms grow exponentially.