Question

Which recursive formula can be used to generate the sequence shown, where f(1) = 5 and n > 1?

5,–1, –7, –13, –19, ...

Answer 1

To determine the recursive formula for the sequence where \( f(1) = 5 \) and the terms are \( 5, -1, -7, -13, -19, \ldots \), let's look at the pattern of the sequence.

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

\[
-1 - 5 = -6
\]
\[
-7 - (-1) = -6
\]
\[
-13 - (-7) = -6
\]
\[
-19 - (-13) = -6
\]

The common difference is \( -6 \). This tells us that each term is obtained by subtracting 6 from the previous term.

Given that the first term \( f(1) \) is 5, we can define the recursive formula as follows:

\[
f(n) = f(n-1) - 6 \quad \text{for } n > 1
\]

We also need to specify the first term:

\[
f(1) = 5
\]

So, the complete recursive formula is:

\[
f(1) = 5
\]
\[
f(n) = f(n-1) - 6 \quad \text{for } n > 1
\]