Question

Use the initial term and the recursive formula to find an explicit formula for the sequence an. Write your answer in simplest form.
a1
=
–
40
an
= 5an–1
an=

Answer 1

To find the explicit formula for the sequence defined by the initial term \( a_1 = -40 \) and the recursive formula \( a_n = 5a_{n-1} \), we start by calculating the first few terms of the sequence:

1. **Calculate the first few terms:**
- \( a_1 = -40 \)
- \( a_2 = 5a_1 = 5(-40) = -200 \)
- \( a_3 = 5a_2 = 5(-200) = -1000 \)
- \( a_4 = 5a_3 = 5(-1000) = -5000 \)
- \( a_5 = 5a_4 = 5(-5000) = -25000 \)

2. **From the pattern:**
We can observe that each term can be expressed as a power of 5 multiplied by the initial term \( a_1 \):
- \( a_2 = 5^1 \cdot a_1 \)
- \( a_3 = 5^2 \cdot a_1 \)
- \( a_4 = 5^3 \cdot a_1 \)
- \( a_5 = 5^4 \cdot a_1 \)

3. **Generalizing this pattern:**
It appears that for any \( n \), we have:
\[
a_n = 5^{n-1} \cdot a_1
\]

4. **Substituting the initial value:**
Since \( a_1 = -40 \), we substitute this into our formula:
\[
a_n = 5^{n-1} \cdot (-40) = -40 \cdot 5^{n-1}
\]

5. **Final explicit formula:**
Thus, the explicit formula for the sequence is:
\[
a_n = -40 \cdot 5^{n-1}
\]

This formula allows you to calculate any term \( a_n \) in the sequence directly without the need for recursion.