Chapter 8.2, Problem 95E

To determine

To find:The change in sum $S_n$ of a finite arithmetic sequence if each term is increased by 5.

Answer to Problem 95E

The sum of new arithmetic sequence will be $S_n+5n$ .

Explanation of Solution

Given information:

A finite arithmetic sequence with nterms , common difference is d , first term $a_1$ and sum is $S_n$ .

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 n .

Common difference is $d$ .

Sum is $S_n$ and first term is $a_1$ .

Last term of the sequence is $a_n$ .

So, relation between first term and last term is

  $a_n=a_1+\left(n-1\right)d$

So, the sum of the finite sequence is calculated as

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

Now, each term is increasing by 5.

So, the sequence can be reformed as

  $a_1+5,a_2+5,a_3+5,\mathrm{...},a_n+5$

Now, the sum of the new sequence will be

  $\begin{array}{c}{S^{\prime }}_n=\frac{n}{2}\left(a_1+5+a_n+5\right)\\ =\frac{n}{2}\left(a_1+a_n+10\right)\\ =\frac{n}{2}\left(a_1+a_n\right)+5n\\ =S_n+5n\end{array}$

Thus, thesum of new arithmetic sequence will be $S_n+5n$ .