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The problem asks to find the explicit function and the 5th term in a sequence that starts with 60 and decreases by 15% each term. Let’s break this down:

### Given:
- The first term \( a_1 = 60 \)
- The sequence decreases by 15% each term (this means each term is 85% of the previous one).

### Recursive Function:
The recursive function for this situation can be written as:
\[
a_n = a_{n-1} \times (1 - 0.15)
\]
or
\[
a_n = a_{n-1} \times 0.85
\]
with \( a_1 = 60 \).

### Explicit Function:
The explicit formula for a geometric sequence is:
\[
a_n = a_1 \times r^{n-1}
\]
where:
- \( a_1 = 60 \) (the first term),
- \( r = 0.85 \) (the common ratio, since the sequence decreases by 15%).

Thus, the explicit function is:
\[
a_n = 60 \times 0.85^{n-1}
\]

### Finding the 5th Term:
To find the 5th term, substitute \( n = 5 \) into the explicit formula:
\[
a_5 = 60 \times 0.85^{4}
\]

Now, let’s calculate this.The 5th term in the sequence is approximately \( 31.3 \) when rounded to the nearest tenth.

### Summary:
- **Explicit function**: \( a_n = 60 \times 0.85^{n-1} \)
- **5th term**: \( 31.3 \)

Would you like further details on this or any clarifications?

Here are five related questions to deepen understanding:
1. What would the 6th term be using the same formula?
2. How does the sequence behave over a longer period, say for the 10th or 20th term?
3. What if the sequence decreased by a different percentage, like 10% or 20%?
4. Can you derive the explicit formula for an arithmetic sequence instead?
5. What is the sum of the first 5 terms of this geometric sequence?

**Tip**: When working with percentage changes in sequences, converting the percentage into a decimal ratio (like 0.85 here) helps simplify both recursive and explicit formulas.

The first term in a sequence is 60. The sequence decreases by 15% each term. Write the recursive function for this situation. What is the 5th term in the sequence (round to the nearest tenths place)? Show your work.

Learn how to solve a geometric sequence problem where the first term is 60 and each subsequent term decreases by 15%. This detailed solution includes both the recursive and explicit formulas, and calculates the 5th term of the sequence, which is approximately 31.3. Suitable for students in Grades 9-10.

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