To determine the 7th term in a geometric sequence, you can 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:

• $a_1 = 5$
• $r = 2$
• $n = 7$

Substitute the values into the formula:

$a_7 = 5 \cdot 2^{(7-1)} = 5 \cdot 2^6$

Calculate $2^6$:

$2^6 = 64$

Now multiply:

$a_7 = 5 \cdot 64 = 320$

So, the 7th term in the geometric sequence is 320.

Would you like further details or have any other questions?

Here are 5 related questions to explore:

1. How do you determine the common ratio in a geometric sequence if not given?
2. What is the sum of the first 7 terms in this geometric sequence?
3. How does the formula change if the sequence is decreasing?
4. What would the 10th term be in this sequence?
5. Can you derive a formula for the sum of the first $n$ terms in a geometric sequence?

Tip: In a geometric sequence, the common ratio is a key factor in determining the nature (increasing or decreasing) and the rate of growth or decay of the sequence.