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
\[f(n)\].
\[f(1)=\]
500
\[f(n)=f(n-1)\]
Related content
\[500\] 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
\[f(n)\].
\[f(1)=\]
500
\[f(n)=f(n-1)\]
Related content
Answers
GPT-4o mini
To analyze the problem, we start with the information provided:
- The initial number of leaves in the first year is \( f(1) = 500 \).
- Each subsequent year, the number of leaves increases by \( 40\% \) compared to the previous year.
This percentage increase indicates that the number of leaves grows by a constant multiplicative factor, specifically \( 1.4 \) (which is \( 100\% + 40\% \)). Therefore, this creates a relationship based on multiplication rather than addition.
Let’s define \( f(n) \) recursively:
- \( f(1) = 500 \)
- For \( n > 1 \):
\[
f(n) = 1.4 \times f(n-1)
\]
This means that each term is obtained by multiplying the previous term by \( 1.4 \).
Based on this examination:
- **Choice A** (Arithmetic sequence) is incorrect because in an arithmetic sequence the difference between consecutive terms is constant, not a multiplicative factor.
- **Choice B** (Geometric sequence) is the correct choice because the ratio between consecutive terms is constant.
To summarize:
1. The sequence \( f(n) \) is a **Geometric sequence**.
2. The complete recursive formula is:
\[
f(1) = 500
\]
\[
f(n) = 1.4 \times f(n-1) \quad \text{for } n > 1
\]
Thus, the answers are:
**(Choice B) Geometric sequence**
And the complete recursive formula is:
\[
f(n) = 1.4 \times f(n-1) \quad \text{for } n > 1
\]
- The initial number of leaves in the first year is \( f(1) = 500 \).
- Each subsequent year, the number of leaves increases by \( 40\% \) compared to the previous year.
This percentage increase indicates that the number of leaves grows by a constant multiplicative factor, specifically \( 1.4 \) (which is \( 100\% + 40\% \)). Therefore, this creates a relationship based on multiplication rather than addition.
Let’s define \( f(n) \) recursively:
- \( f(1) = 500 \)
- For \( n > 1 \):
\[
f(n) = 1.4 \times f(n-1)
\]
This means that each term is obtained by multiplying the previous term by \( 1.4 \).
Based on this examination:
- **Choice A** (Arithmetic sequence) is incorrect because in an arithmetic sequence the difference between consecutive terms is constant, not a multiplicative factor.
- **Choice B** (Geometric sequence) is the correct choice because the ratio between consecutive terms is constant.
To summarize:
1. The sequence \( f(n) \) is a **Geometric sequence**.
2. The complete recursive formula is:
\[
f(1) = 500
\]
\[
f(n) = 1.4 \times f(n-1) \quad \text{for } n > 1
\]
Thus, the answers are:
**(Choice B) Geometric sequence**
And the complete recursive formula is:
\[
f(n) = 1.4 \times f(n-1) \quad \text{for } n > 1
\]