Question
Which recursive formula can be used to define this sequence for n>1? 15, 22, 29, 36, 43, 50, ...
Answers
GPT-4o mini
To define the sequence \( 15, 22, 29, 36, 43, 50, \ldots \) recursively, we first need to identify the pattern of how the terms are generated.
Let's examine the differences between consecutive terms:
- \( 22 - 15 = 7 \)
- \( 29 - 22 = 7 \)
- \( 36 - 29 = 7 \)
- \( 43 - 36 = 7 \)
- \( 50 - 43 = 7 \)
The difference between each consecutive term is consistently \( 7 \). This indicates that the sequence is an arithmetic sequence with a common difference of \( 7 \).
Based on this observation, we can define the sequence recursively as follows:
- Let \( a_1 = 15 \) (the first term).
- For \( n > 1 \):
\[
a_n = a_{n-1} + 7
\]
So, the recursive formula for the given sequence is:
\[
\begin{align*}
a_1 &= 15 \\
a_n &= a_{n-1} + 7 \quad \text{for } n > 1
\end{align*}
\]
Let's examine the differences between consecutive terms:
- \( 22 - 15 = 7 \)
- \( 29 - 22 = 7 \)
- \( 36 - 29 = 7 \)
- \( 43 - 36 = 7 \)
- \( 50 - 43 = 7 \)
The difference between each consecutive term is consistently \( 7 \). This indicates that the sequence is an arithmetic sequence with a common difference of \( 7 \).
Based on this observation, we can define the sequence recursively as follows:
- Let \( a_1 = 15 \) (the first term).
- For \( n > 1 \):
\[
a_n = a_{n-1} + 7
\]
So, the recursive formula for the given sequence is:
\[
\begin{align*}
a_1 &= 15 \\
a_n &= a_{n-1} + 7 \quad \text{for } n > 1
\end{align*}
\]