Question

Write a recursive formula for an
​, the n^th
term of the sequence 8,2,−4,−10,...(1 point)
an=an−1−

Answer 1

To find a recursive formula for the sequence \( a_n \) which starts with the terms 8, 2, -4, -10, ..., let's first identify the pattern between the terms.

1. **Identify the first term:**
- \( a_1 = 8 \)

2. **Calculate the differences between the terms:**
- \( a_2 - a_1 = 2 - 8 = -6 \)
- \( a_3 - a_2 = -4 - 2 = -6 \)
- \( a_4 - a_3 = -10 - (-4) = -6 \)

From this observation, we can see that each term is obtained by subtracting 6 from the previous term.

3. **Write the recursive formula:**
The recursive formula for the sequence can be stated as:
\[
a_n = a_{n-1} - 6
\]
with the initial condition:
\[
a_1 = 8
\]

So, the complete recursive formula is:
\[
a_n = a_{n-1} - 6 \quad \text{for } n \geq 2, \quad a_1 = 8
\]