What is the common ratio in a geometric sequence if the first term is 10 and the fifth term is 160?

A. The common ratio in the sequence is 40.

B. The common ratio in the sequence is 0.25.

C. The common ratio in the sequence is 16.

D. The common ratio in the sequence is 2.

To find the common ratio \( r \) in a geometric sequence, we can use the formula for the \( n \)-th term of a geometric sequence:

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

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:
- The first term \( a_1 = 10 \),
- The fifth term \( a_5 = 160 \),
- \( n = 5 \).

We substitute these into the formula for the fifth term:

\[
160 = 10 \times r^{5-1}
\]

Simplify the exponent:

\[
160 = 10 \times r^4
\]

Next, solve for \( r^4 \) by dividing both sides by 10:

\[
r^4 = \frac{160}{10} = 16
\]

To find \( r \), take the fourth root of both sides:

\[
r = \sqrt[4]{16} = 2
\]

Therefore, the common ratio is \( r = 2 \).

The correct answer is **D. The common ratio in the sequence is 2**.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you find the nth term of a geometric sequence?
2. What is the sum of the first \( n \) terms in a geometric sequence?
3. How does the common ratio affect the growth of a geometric sequence?
4. Can the common ratio in a geometric sequence be negative? What would that look like?
5. How do geometric sequences differ from arithmetic sequences?

**Tip:** Always check if the common ratio is consistent by verifying it with another term in the sequence!

Learn how to find the common ratio in a geometric sequence where the first term is 10 and the fifth term is 160. Step-by-step solution provided with formula application and calculation. Suitable for Grades 9-12 mathematics students.

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