Question

Solve : $25+22+19+16+....+x=115$

Answer 1

$25+23+19+16+...+x=115$

Its an A.P., where

$a=25d=22-25=(-3),S_n=115$

in order to find $x$, we need to find the number of terms i.e. $n$

$S_n=\frac{n}{2}(2a+(n-1\left)d\right)$

$115=\frac{n}{2}(2\times 25+(n-1\left)\right(-3\left)\right)$

$115=\frac{n}{2}(50+n(-3)-(-3\left)\right)$

$115=\frac{n}{2}(50-3n+3)$

$115=\frac{n}{2}(53-3n)$

$115\times 2=n(53-3n)$

$230=53n-3n^2$

$230-53n+3n^2=0$

Here we have a quadratic equation i.e.

$3n^2-53n+230=0$

to find the value of n, we need to find the roots of the equation by the formula,

$n=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

where $a=3,b=-53andc=230$

$n=\frac{-(-53)\pm \sqrt{(-53{)}^2-4\times 3\times 230}}{2\times 3}$

$n=\frac{53\pm \sqrt{2809-2760}}{6}$

$n=\frac{53\pm 7}{6}$

$n=\frac{53+7}{6}$ or $n=\frac{53-7}{6}$

$n=\frac{60}{6}$ or $n=\frac{46}{6}$

$n=10$ or $n=7.66$

Since the number of terms i.e. $n$ cannot be a decimal number

Therefore $n=10$

for finding the last term,

$x=a+(n-1)d$

$x=25+(10-1)(-3)$

$x=25+9(-3)$

$x=25-27$

$x=-2$

So, the last term of the series i.e. $x$ is $-2$