Question
Write a recursive formula for the following explicit formula, and find the first term of the sequence. \[ a_{n}=31+41 \mathrm{n} \] \( a_{1}=\square \) \( a_{n}=\square \) (Simplify your answers.)
Ask by Turner Riley. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
First term: \( a_1 = 72 \)
Recursive formula:
\[
a_n = a_{n-1} + 41 \quad \text{for } n > 1
\]
Solution
To convert the explicit formula \( a_n = 31 + 41n \) into a recursive formula and find the first term of the sequence, follow these steps:
1. **Find the first term (\( a_1 \))**:
\[
a_1 = 31 + 41 \times 1 = 31 + 41 = 72
\]
2. **Determine the recursive relationship**:
Notice that each term increases by 41 from the previous term. Therefore, the recursive formula can be written as:
\[
a_n = a_{n-1} + 41 \quad \text{for } n > 1
\]
**Final Recursive Formula:**
\[
\begin{cases}
a_1 = 72 \\
a_n = a_{n-1} + 41 & \text{for } n > 1
\end{cases}
\]
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Simplify this solution Mind Expander
To find the first term of the sequence defined by the explicit formula \( a_{n} = 31 + 41n \), we simply substitute \( n = 1 \): \[ a_{1} = 31 + 41(1) = 31 + 41 = 72. \] Now, let's derive the recursive formula. The recursive formula expresses each term in relation to the previous term. Noticing that the difference between consecutive terms \( a_{n} \) can be represented as: \[ a_{n} - a_{n-1} = (31 + 41n) - (31 + 41(n-1)) = 41. \] Thus, we can express the recursive formula as follows: \[ a_{n} = a_{n-1} + 41 \text{ for } n \geq 2 \] with the initial condition: \[ a_{1} = 72. \] So, filling in the blanks: \( a_{1}=72 \) \( a_{n}=a_{n-1}+41 \)
