Question

Julia owns 13 books. She buys two new books every week. Which arithmetic sequence describes the number of books she has?
A. a1 = 13; an = an − 1 + 2
B. a1 = 2; an= an − 1 + 13
C. a1 = 13; an = an + 1 + 2
D. a1 = 2; an= an +1+13

Answer 1

To describe the number of books Julia has over time as she buys two new books every week, we can set up an arithmetic sequence.

- Julia starts with 13 books, so \( a_1 = 13 \).
- Each week, she buys 2 more books, which means the common difference \( d = 2 \).

The general formula for the \( n \)-th term of an arithmetic sequence is given by:

\[ a_n = a_1 + (n - 1) \cdot d \]

In her scenario, the sequence can also be expressed in a recursive form as:

\[ a_n = a_{n - 1} + 2 \]

Looking at the answer choices:

- **A.** \( a_1 = 13; a_n = a_{n - 1} + 2 \) (This is correct.)
- **B.** \( a_1 = 2; a_n = a_{n - 1} + 13 \) (This is incorrect; starts counting wrong and adds wrong amount.)
- **C.** \( a_1 = 13; a_n = a_n + 1 + 2 \) (This is incorrect; wrong formula structure.)
- **D.** \( a_1 = 2; a_n = a_n + 1 + 13 \) (This is incorrect; starts counting wrong and wrong formula structure.)

The correct answer is **A.** \( a_1 = 13; a_n = a_{n - 1} + 2 \).