Question

How do you find the ${25}^{th}$partial sum of the arithmetic sequence ${\mathrm{a}}_1=100,{\mathrm{a}}_{25}=220$?

Answer 1

Find the required partial sum:

Given : ${\mathrm{a}}_1=100,{\mathrm{a}}_{25}=220$

In an arithmetic sequence, sum of $\mathrm{n}$ terms of series is

${\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}\left[{\mathrm{a}}_1+{\mathrm{a}}_{\mathrm{l}}\right]$, where ${\mathrm{a}}_1$ is the first term and ${\mathrm{a}}_{\mathrm{l}}$ is the last term.

Now, to find the sum of $25$ terms of arithmetic sequence, substitute $\mathrm{n}=25$

${\mathrm{S}}_{25}=\frac{25}{2}\left[{\mathrm{a}}_1+{\mathrm{a}}_{25}\right]$

$\Rightarrow$ ${\mathrm{S}}_{25}=\frac{25}{2}\left[100+220\right]$

$\Rightarrow$ ${\mathrm{S}}_{25}=\frac{25}{2}\left[320\right]$

$\Rightarrow$ ${\mathrm{S}}_{25}=25\times 160$

$\Rightarrow$ ${\mathrm{S}}_{25}=4000$

Hence, the ${25}^{th}$partial sum of the given arithmetic sequence is $4000$.