Question

3 julia owns 13 books. she buys two new books every week. which arithmetic sequence describes the number of books she has? a $a_1 = 13$; $a_n = a_{n - 1} + 2$ b $a_1 = 2$; $a_n = a_{n - 1} + 13$ c $a_1 = 13$; $a_n = a_{n + 1} + 2$ d $a_1 = 2$; $a_n = a_{n + 1} + 13$

Explanation:

Step1: Analyze the initial term

Julia starts with 13 books, so the first term \(a_1\) of the sequence should be 13. This eliminates options B and D where \(a_1 = 2\).

Step2: Analyze the recursive formula

She buys 2 new books every week, meaning each subsequent term (number of books in the next week) is 2 more than the previous term. In a recursive formula for an arithmetic sequence, \(a_n=a_{n - 1}+d\) where \(d\) is the common difference. Here, \(d = 2\), so the formula should be \(a_n=a_{n-1}+2\). Option C has \(a_n=a_{n + 1}+2\) which would imply the number of books is decreasing by 2, which is incorrect.

Answer:

A. \(a_1 = 13\); \(a_n = a_{n-1}+2\)