Question

Find $x,$ if $25+22+19+......+x=115$.

Answer 1

Consider the given series in A.P.

$25+22+19+............+x=115$

Then,

$a=25$

$d=22-25=-3$

$S_n=115$

$n=?$

We know that,

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

$x=25+(n-1)\times (-3)$

$x=25-3n+3$

$x=28-3n...........\left(1\right)$

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

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

$\Rightarrow 230=n(50-3n+3)$

$\Rightarrow 230=53n-3n^2$

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

$\Rightarrow 3n^2-(30+23)n+230=0$

$\Rightarrow 3n^2-30n-23n+230=0$

$\Rightarrow 3n(n-10)-23(n-10)=0$

$\Rightarrow (n-10)(3n-23)=0$

If

$n-10=0$

$n=10$

$3n-23=0$

$n=\frac{23}{3}$

$n=10$ is acceptable

Put in the equation (1) and we get,

$x=28-3n$

$x=28-3\times 10$

$x=-2$

Hence, this is the answer.