Chapter 8.2, Problem 63E

To calculate: The ${25}^{th}$ partial sum of the arithmetic sequence.

To determine

The ${25}^{th}$ partial sum of the arithmetic sequenceis $4000$ .

Given information:

The given sequence has $a_1=100,a_{25}=100$

Definition used:

The 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.

The sum of finite arithmetic sequence is given by $S_n=\frac{n\left(a_1+a_n\right)}{2}$ .

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

Calculation:

It is given that $a_1=100,a_{25}=100$ .

Compute the 25^th partial sum as follows,

  $\begin{array}{c}{S}_{25}=\frac{25\left(100+220\right)}{2}\\ =25\cdot 160\\ =4000\end{array}$

Therefore, the sum of the partial arithmetic sequence is $4000$ .

Answer to Problem 63E

The ${25}^{th}$ partial sum of the arithmetic sequenceis $4000$ .

Explanation of Solution

Given information:

The given sequence has $a_1=100,a_{25}=100$

Definition used:

The 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.

The sum of finite arithmetic sequence is given by $S_n=\frac{n\left(a_1+a_n\right)}{2}$ .

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

Calculation:

It is given that $a_1=100,a_{25}=100$ .

Compute the 25^th partial sum as follows,

  $\begin{array}{c}{S}_{25}=\frac{25\left(100+220\right)}{2}\\ =25\cdot 160\\ =4000\end{array}$

Therefore, the sum of the partial arithmetic sequence is $4000$ .