Question
Write a recursive formula for the following explicit formula, and find the first term of \( t \) sequence. \( a_{n}=31+41 \mathrm{n} \) \( a_{1}=\square \) \( a_{n}=\square \) (Simplify your answers.)
Ask by Gough Owen. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
First term \( a_1 = 72 \).
Recursive formula: \( a_n = a_{n-1} + 41 \).
Solution
To convert the explicit formula \( a_n = 31 + 41n \) into a recursive formula and find the first term \( a_1 \), follow these steps:
1. **Find the First Term (\( a_1 \)):**
Substitute \( n = 1 \) into the explicit formula:
\[
a_1 = 31 + 41 \times 1 = 31 + 41 = 72
\]
2. **Derive the Recursive Formula:**
A recursive formula defines each term based on the previous term. To find the relationship between \( a_n \) and \( a_{n-1} \):
\[
a_n = 31 + 41n
\]
\[
a_{n-1} = 31 + 41(n-1) = 31 + 41n - 41 = 41n - 10
\]
Subtract the two equations to find the difference:
\[
a_n - a_{n-1} = (31 + 41n) - (41n - 10) = 31 + 10 = 41
\]
This shows that each term increases by 41 from the previous term. Therefore, the recursive formula is:
\[
a_n = a_{n-1} + 41 \quad \text{for } n \geq 2
\]
**Final Answer:**
\[
a_1 = 72
\]
\[
a_n = a_{n-1} + 41
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
Let's break it down! The explicit formula given is \( a_{n} = 31 + 41n \). To find the first term of the sequence, simply plug in 1 for \( n \): \[ a_{1} = 31 + 41(1) = 31 + 41 = 72. \] So, \( a_{1} = 72 \). Next, for the recursive formula, you need to express it in terms of the previous term. We notice that the difference between consecutive terms is 41. This leads us to: \[ a_{n} = a_{n-1} + 41 \] where \( a_{1} = 72 \). So, the answers are: \( a_{1} = 72 \) \( a_{n} = a_{n-1} + 41 \)
