# Solve the equation$-4+(-1)+2+\ldots+x=437$

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Given:

$(-4) + (-1) + 2 + 5 + … + x = 437$.

To do:

We have to find the value of $x$.

Solution:

Given equation is $(-4)+(-1)+2+5+\ldots+x=437 \quad \ldots(i)$

LHS: $(-4), (-1), 2, 5, \ldots x$

This forms an A.P., where

First term $a=-4,$ common difference $=(-1)-(-4)=-1+4=3$,

$a_{n}=l=x$

We know that,

$n$ th term of an A.P., $a_{n}=l=a+(n-1) d$

$\Rightarrow x=-4+(n-1) 3$

$\Rightarrow \frac{x+4}{3}=n-1$

$\Rightarrow n=\frac{x+7}{3}$

We know that,

Sum of an A.P., $S_{n}=\frac{n}{2}[2 a+(n-1) d]$

$S_{n}=\frac{x+7}{2 \times 3}[2(-4)+(\frac{x+4}{3})3]$

$=\frac{x+7}{2 \times 3} (-8+x+4)$

$=\frac{(x+7)(x-4)}{2 \times 3}$

$\Rightarrow \frac{(x+7)(x-4)}{2 \times 3}=437$                (From (i))

$\Rightarrow x^{2}+7 x-4 x-28=874 \times 3$

$\Rightarrow x^{2}+3 x-2650=0$

$x=\frac{-3 \pm \sqrt{(3)^{2}-4(-2650)}}{2}$

$x=\frac{-3 \pm \sqrt{9+10600}}{2}$

$x=\frac{-3 \pm \sqrt{10609}}{2}$

$x=\frac{-3 \pm 103}{2}$

$x=\frac{100}{2}$ or $\frac{-106}{2}$

$x=50$ or $x=-53$ which is not possible, for $x=-53, n$ will be negative which is not possible.

Hence, the required value of $x$ is 50.

Updated on 10-Oct-2022 13:27:44