Question

Write a recursive formula for a, start subscript, n, end subscripta
n
​
, the n, start superscript, th, end superscriptn
th
term of the sequence 16, comma, 7, comma, minus, 2, comma, minus, 11, comma, dot dot dot16,7,−2,−11,....
Answer
Attempt 1 out of 2

a, start subscript, 1, end subscript, equalsa
1
​
=

a, start subscript, n, end subscript, equalsa
n
​
=

Answer 1

To find a recursive formula for the sequence given \( 16, 7, -2, -11, \ldots \), let's observe the pattern between the terms.

1. **Determine the relationship between the terms**:
- \( a_1 = 16 \)
- \( a_2 = 7 \) (difference from \( a_1 \): \( 7 - 16 = -9 \))
- \( a_3 = -2 \) (difference from \( a_2 \): \( -2 - 7 = -9 \))
- \( a_4 = -11 \) (difference from \( a_3 \): \( -11 - (-2) = -9 \))

From this observation, we can see that each term is obtained by subtracting 9 from the previous term.

2. **Formulate the recursive relation**:
- Base case: \( a_1 = 16 \)
- Recursive case: \( a_n = a_{n-1} - 9 \) for \( n \geq 2 \)

Thus, the recursive formula for the sequence is:

\[
\begin{align*}
a_1 & = 16, \\
a_n & = a_{n-1} - 9 \quad \text{for } n \geq 2.
\end{align*}
\]