Write a recursive formula for the following explicit formula, and find the first term of the
sequence.

(Simplify your answers.)

Ask by Wheeler Rodriguez. in the United States

Jan 23,2025

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.)

UpStudy AI · Accepted Answer

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 $$ a_1 $$:**
   $$
   a_1 = 31 + 41(1) = 31 + 41 = 72
   $$

2. **Derive the recursive formula:**
   
   A recursive formula expresses $$ a_n $$ in terms of the previous term $$ a_{n-1} $$. To find this relationship:
   $$
   a_n = 31 + 41n
   $$
   $$
   a_{n-1} = 31 + 41(n-1) = 31 + 41n - 41 = (31 - 41) + 41n = -10 + 41n
   $$
   
   Subtract $$ a_{n-1} $$ from $$ a_n $$:
   $$
   a_n - a_{n-1} = (31 + 41n) - (-10 + 41n) = 31 + 41n + 10 - 41n = 41
   $$
   
   Therefore, the recursive formula is:
   $$
   a_n = a_{n-1} + 41
   $$

**Final Answer:**
$$
a_1 = 72, \quad a_n = a_{n-1} + 41
$$
First term: 72 Recursive formula: $$ a_n = a_{n-1} + 41 $$

Write a recursive formula for the following explicit formula, and find the first term of the
sequence.

(Simplify your answers.)

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Write a recursive formula for the following explicit formula, and find the first term of the sequence. (Simplify your answers.)