Write a recursive formula for the sequence left-brace a subscript n baseline right-brace equals left-brace 15 comma 26 comma 37 comma 48 comma 59 comma ellipsis right-brace.

Question

Write a recursive formula for the sequence left-brace a subscript n baseline right-brace equals left-brace 15 comma 26 comma 37 comma 48 comma 59 comma ellipsis right-brace.
(1 point)
Responses

a subscript 1 baseline equals 15, 
Image with alt text: a subscript 1 baseline equals 15 ,  Image with alt text:

GPT-4o mini · Answer

To derive a recursive formula for the sequence \( { a_n } = { 15, 26, 37, 48, 59, \ldots } \), we first observe that the difference between consecutive terms is constant.

Calculating the differences:

\( a_2 - a_1 = 26 - 15 = 11 \)
\( a_3 - a_2 = 37 - 26 = 11 \)
\( a_4 - a_3 = 48 - 37 = 11 \)
\( a_5 - a_4 = 59 - 48 = 11 \)

The common difference is \( 11 \). This indicates that the recursive formula can be constructed using this difference.

A suitable recursive formula for this sequence is:

\[ \begin{cases} a_1 = 15 \ a_n = a_{n-1} + 11 \quad \text{for } n > 1 \end{cases} \]

This formula states that the first term \( a_1 \) is 15, and each subsequent term \( a_n \) is obtained by adding 11 to the previous term \( a_{n-1} \).