Question

Which recursive formula can be used to generate the sequence below, where f(1) = 6 and n ≥ 1?

6, 1, –4, –9, –14, …

Answer 1

[AI-generated answer — Bot GPT-4o]

To derive a recursive formula for the sequence \(6, 1, -4, -9, -14, \ldots\), where \(f(1) = 6\) and \(n \geq 1\), we need to determine how each term relates to its predecessor.

Given the sequence:
\[ 6, 1, -4, -9, -14, \ldots \]

First, let's find the difference between consecutive terms:

\[ 1 - 6 = -5 \]
\[ -4 - 1 = -5 \]
\[ -9 - (-4) = -5 \]
\[ -14 - (-9) = -5 \]

We observe that the difference between any two consecutive terms is constant and equal to \(-5\). Therefore, the sequence is an arithmetic sequence with a common difference of \(-5\).

The recursive formula for an arithmetic sequence with a first term \(f(1)\) and common difference \(d\) is given by:
\[ f(n) = f(n-1) + d \]

For this sequence:
- The first term \(f(1) = 6\)
- The common difference \(d = -5\)

Thus, the recursive formula is:
\[ f(1) = 6 \]
\[ f(n) = f(n-1) - 5 \quad \text{for } n > 1 \]

So, the recursive formula for generating this sequence is:
\[ f(1) = 6 \]
\[ f(n) = f(n-1) - 5 \quad \text{for } n \geq 2 \]