Question
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. \( a_{1}=2 ; a_{n}=a_{n-1}+13 \) D. \( a_{1}=13 ; a_{n}=a_{n-1}+2 \)
Ask by Turnbull Colon. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The correct arithmetic sequence is \( a_{1}=13 ; a_{n}=a_{n-1}+2 \).
Solution
To determine the correct arithmetic sequence that describes the number of books Julia has, let's break down the information:
1. **Initial Number of Books**: Julia starts with 13 books. This means the first term of the sequence, \( a_1 \), is 13.
2. **Weekly Addition**: Julia buys 2 new books every week. In an arithmetic sequence, the common difference (\( d \)) is the constant amount added each term. Here, \( d = 2 \).
3. **General Formula**: The \( n \)-th term of an arithmetic sequence can be described by the formula:
\[
a_n = a_{n-1} + d
\]
Substituting the known values:
\[
a_n = a_{n-1} + 2
\]
Putting it all together, the correct arithmetic sequence is:
\[
a_1 = 13; \quad a_n = a_{n-1} + 2
\]
**Answer:**
**D.** \( a_{1}=13 ; a_{n}=a_{n-1}+2 \)
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Simplify this solution Bonus Knowledge
The correct choice is D. In this arithmetic sequence, Julia starts with \( a_{1} = 13 \) books. Each week, she adds 2 new books, so the next term \( a_{n} \) is obtained by taking the previous term \( a_{n-1} \) and adding 2, forming the pattern that describes her growing book collection. Did you know that arithmetic sequences can be traced back to ancient civilizations, with early mathematicians using them for everything from astronomy to trade? They laid the groundwork for our understanding of patterns in numbers, making their discoveries timeless treasures that still apply today!
