Question

Mohamed decided to track the number of leaves on the tree in his backyard each year. The first year, there were
\[500\] leaves. Each year thereafter, the number of leaves was
\[40\%\] more than the year before.
Let
\[f(n)\] be the number of leaves on the tree in Mohamed's back yard in the
\[n^\text{th}\] year since he started tracking it.
\[f\] is a sequence. What kind of sequence is it?
Choose 1 answer:
Choose 1 answer:
(Choice A) Arithmetic sequence
A
Arithmetic sequence
(Choice B) Geometric sequence
B
Geometric sequence
Complete the recursive formula for
\[f(n)\].
\[f(1)=\]
\[f(n)=f(n-1)\]

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Answer 1

The sequence \( f(n) \) represents the number of leaves each year, where each year's count is \( 40\% \) more than the previous year's count. This means that the growth is multiplicative rather than additive. Specifically, each year, the count of leaves is multiplied by a factor of \( 1.4 \) (since \( 100\% + 40\% = 140\% \) or \( 1.4 \)).

This makes the sequence a **Geometric sequence**.

To complete the recursive formula:

1. We know the initial value:
\[
f(1) = 500
\]

2. For subsequent years, we express it using the previous year's count:
\[
f(n) = 1.4 \cdot f(n-1)
\]

Thus, the completed recursive formula is:

\[ f(1) = 500 \]
\[ f(n) = 1.4 \cdot f(n-1) \]