Chapter 8.2, Problem 86E

To determine

To find:The first term of the arithmetic sequence.

Answer to Problem 86E

The first term is $4$ .

Explanation of Solution

Given information:

A finite arithmetic sequence with 20 terms, common difference is 3 and sum is 650.

Concept used:

An arithmetic sequence of n terms, has the form $a_1,a_2,a_3,\mathrm{...},a_n$ and should have a common difference between each two consecutive terms.

That is

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}$

Common difference can be defined by d .

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}=d$

nth term of the arithmetic sequence has the form

  $a_n=a_1+\left(n-1\right)d$

Where $a_1$ is the first term of the sequence, and d is the common difference.

Sum of an arithmetic finite sequence has the form

  $S_n=\frac{n}{2}\left(a_1+a_n\right)$

Here, n is number of terms, $a_1$ is the first term of the sequence, and $a_n$ is the last term of the sequence.

Calculation:

Number of terms is 20.

  $n=20$

Common difference is

  $d=3$

Now, consider first term of the finite sequence is $a_1$ .

Last term of the sequence is $a_n$ .

So, relation between first term and last term is

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ a_n=a_1+\left(20-1\right)3\\ a_n=a_1+57\end{array}$

So, the sum of the finite sequence is calculated as

  $\begin{array}{c}{S}_n=\frac{n}{2}\left(a_1+a_n\right)\\ S_{20}=\frac{20}{2}\left(a_1+a_n\right)\\ 650=10\times \left(a_1+a_n\right)\\ 650=10\times \left(a_1+a_1+57\right)\\ 650=10\left(2a_1+57\right)\\ 650=20a_1+570\\ 20a_1=650-570\\ 20a_1=80\\ a_1=4\end{array}$

Thus, the first term is $4$ .