Question
Which recursive formula can be used to define this sequence for n>1? 6, 11, 16, 21, 26, 31, ...
Answers
GPT-4o mini
To define the sequence \( 6, 11, 16, 21, 26, 31, \ldots \) recursively, we first need to identify the pattern in the sequence.
The difference between successive terms is constant. Specifically:
- \( 11 - 6 = 5 \)
- \( 16 - 11 = 5 \)
- \( 21 - 16 = 5 \)
- \( 26 - 21 = 5 \)
- \( 31 - 26 = 5 \)
This shows that each term is obtained by adding 5 to the previous term.
To express this recursively, we can denote the sequence as \( a_n \), where \( n \) is the term index. The recursive formula for this sequence, starting from \( n > 1 \), can be written as follows:
\[
a_1 = 6 \quad (initial\ term)
\]
\[
a_n = a_{n-1} + 5 \quad (for\ n > 1)
\]
So the recursive formula for the given sequence is:
\[
\begin{align*}
a_1 &= 6 \\
a_n &= a_{n-1} + 5 \quad (n > 1)
\end{align*}
\]
The difference between successive terms is constant. Specifically:
- \( 11 - 6 = 5 \)
- \( 16 - 11 = 5 \)
- \( 21 - 16 = 5 \)
- \( 26 - 21 = 5 \)
- \( 31 - 26 = 5 \)
This shows that each term is obtained by adding 5 to the previous term.
To express this recursively, we can denote the sequence as \( a_n \), where \( n \) is the term index. The recursive formula for this sequence, starting from \( n > 1 \), can be written as follows:
\[
a_1 = 6 \quad (initial\ term)
\]
\[
a_n = a_{n-1} + 5 \quad (for\ n > 1)
\]
So the recursive formula for the given sequence is:
\[
\begin{align*}
a_1 &= 6 \\
a_n &= a_{n-1} + 5 \quad (n > 1)
\end{align*}
\]