determine 7th term in geometric sequence whose first term is 5 and whose common ratio is 2

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.

Learn how to calculate the 7th term in a geometric sequence with a first term of 5 and a common ratio of 2 using the geometric sequence nth term formula. Understand the principles of geometric sequences and apply them to solve problems. Suitable for high school students and beyond.

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