Question

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

6, 1, –4, –9, –14, …

f (n + 1) = f(n) + 5
f (n + 1) = f(n) – 5
f (n) = f(n + 1 ) – 5
f (n + 1) = –5f(n)

Answer 1

Three terms of an arithmetic sequence are shown below. Which recursive formula defines the sequence?

f(1) = 6, f(4) = 12, f(7) = 18

f (n + 1) = f(n) + 6
f (n + 1) = 2f(n)
f (n + 1) = f(n) + 2
f (n + 1) = 1.5f(n)

Answer 2

To generate the sequence 6, 1, –4, –9, –14, … where \( f(1) = 6 \) and \( n \geq 1 \), we need to identify a recursive formula that matches the sequence. Let’s analyze the differences between consecutive terms in the sequence:

\[
\begin{align*}
f(1) & = 6, \\
f(2) & = 1, \\
f(3) & = -4, \\
f(4) & = -9, \\
f(5) & = -14, \\
\end{align*}
\]

Now, calculate the differences:

\[
\begin{align*}
f(2) - f(1) & = 1 - 6 = -5, \\
f(3) - f(2) & = -4 - 1 = -5, \\
f(4) - f(3) & = -9 - (-4) = -5, \\
f(5) - f(4) & = -14 - (-9) = -5.
\end{align*}
\]

Each of these differences is \( -5 \). This shows that each term in the sequence is obtained by subtracting \( 5 \) from the previous term.

Therefore, the recursive formula that can be used to generate the sequence is:

\[ f(n + 1) = f(n) - 5 \]

This matches the second option in the given choices:

\[ f(n + 1) = f(n) - 5 \]