Chapter 8.2, Problem 69E

To determine

To find:The25^th partial sum of given arithmetic sequence.

Answer to Problem 69E

The 25^th partial sum of the given arithmetic sequence is $4000$ .

Explanation of Solution

Given information:

An arithmetic sequence is given as $a_1=100,a_{25}=100$ .

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:

Consider the given sequence.

  $a_1=100,a_{25}=100$

Number of term for the required sumis indicated as 25.

So, the 25^th partial sum is calculated as

  $\begin{array}{c}{S}_n=\frac{n}{2}\left(a_1+a_n\right)\\ S_{25}=\frac{25}{2}\left(100+220\right)\\ =25\times 160\\ =4000\end{array}$

Thus, the 25^th partial sum of the given arithmetic sequence is $4000$ .