Aiko listed a geometric in her paper. The first term is 10 while the last term is 5/32. How many numbers did she write if she multiplied 1/2 by every term to get the next term?

To determine the number of terms in Aiko's geometric sequence, we need to use the properties of geometric sequences. 

In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio. Let's denote:
- \( a \) as the first term of the sequence,
- \( r \) as the common ratio,
- \( l \) as the last term of the sequence,
- \( n \) as the number of terms in the sequence.

Given:
- The first term \( a = 10 \),
- The last term \( l = \frac{5}{32} \),
- The common ratio \( r = \frac{1}{2} \).

The \( n \)-th term of a geometric sequence can be calculated using the formula:
\[ a_n = a \cdot r^{n-1} \]

We know \( a_n = l \). Thus:
\[ \frac{5}{32} = 10 \cdot \left( \frac{1}{2} \right)^{n-1} \]

To solve for \( n \), we first isolate \( \left( \frac{1}{2} \right)^{n-1} \):
\[ \frac{5}{32} = 10 \cdot \left( \frac{1}{2} \right)^{n-1} \]
\[ \left( \frac{1}{2} \right)^{n-1} = \frac{5}{32} \div 10 \]
\[ \left( \frac{1}{2} \right)^{n-1} = \frac{5}{32} \cdot \frac{1}{10} \]
\[ \left( \frac{1}{2} \right)^{n-1} = \frac{5}{320} \]
\[ \left( \frac{1}{2} \right)^{n-1} = \frac{1}{64} \]

Since \( \left( \frac{1}{2} \right)^6 = \frac{1}{64} \), we can equate:
\[ n-1 = 6 \]
\[ n = 7 \]

Therefore, Aiko wrote **7** numbers in her geometric sequence.

Would you like more details or have any questions? Here are some related questions:

1. What is a geometric sequence?
2. How do you determine the common ratio in a geometric sequence?
3. What is the general formula for the \( n \)-th term of a geometric sequence?
4. How can you find the sum of the first \( n \) terms of a geometric sequence?
5. How do you determine if a sequence is geometric?
6. Can geometric sequences have negative common ratios?
7. What happens if the common ratio in a geometric sequence is greater than 1?
8. How do you derive the formula for the sum of an infinite geometric series?

**Tip:** When solving for \( n \) in geometric sequences, always check if your answer makes sense by plugging back into the original sequence formula.

Learn how to calculate the number of terms in a geometric sequence where the first term is 10 and the last term is 5/32, with a common ratio of 1/2. This problem involves using the formula for the n-th term of a geometric sequence. Suitable for students in Grades 9-12.

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