Question

1. each year, a volunteer organization expects to add 5 more people for whom the group provides home maintenance services. this year, the organization provides the service for 32 people.

a. write an explicit formula for the number of people the organization expects to serve each year.
b. how many people would the organization expect to serve during the year, 20 years from now?

of an arithmetic series with the given number of terms, a₁ and aₙ.

Explanation:

Part (a)

Step 1: Identify the sequence type

This is an arithmetic sequence where the first term \(a_1 = 32\) (people served this year) and the common difference \(d = 5\) (additional people each year). The explicit formula for the \(n\)-th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\).

Step 2: Substitute values into the formula

Substitute \(a_1 = 32\) and \(d = 5\) into the formula:
\(a_n=32+(n - 1)\times5\)
Simplify the formula:
\(a_n=32 + 5n-5=27 + 5n\)

Step 1: Determine the value of \(n\)

We want to find the number of people served 20 years from now, so \(n=20 + 1=21\) (since \(n = 1\) is this year).

Step 2: Use the explicit formula

From part (a), the formula is \(a_n=5n + 27\). Substitute \(n = 21\) into the formula:
\(a_{21}=5\times21+27\)

Step 3: Calculate the value

First, calculate \(5\times21 = 105\). Then, \(105+27=132\).

Answer:

The explicit formula is \(a_n = 5n+27\) (where \(n\) is the number of years from now, \(n = 1\) for this year)

Part (b)