Question

Question 8
Solve the equation -4 + (-1) + 2 + . . . . . + x = 437.

Answer 1

Given equation is, - 4 - 1 + 2 + . . . . .+ x = 437 . . . .. (i)
Here, - 4 - 1 + 2 + . . . . .+ x forms an AP with first term= -4, common difference = 3,
$a_n$ = l = x
$\because$ nth term of an AP, $a_n=l=a+(n-1)d$
$\Rightarrow x=-4+(n-1)3$
$\Rightarrow \frac{x+4}{3}=n-1\Rightarrow n=\frac{x+7}{3}$
$\therefore SumofanAP,S_n=\frac{n}{2}[2a+(n-1\left)d\right]$
$S_n=\frac{x+7}{2\times 3}\left[2\right(-4)+\left(\frac{x+4}{3}\right).3]$
$=\frac{x+7}{2\times 3}(-8+x+4)=\frac{(x+7)(x-4)}{2\times 3}$
From Eq.(i),
$S_n$ = 437
$\Rightarrow \frac{(x+7)(x-4)}{2\times 3}=437$
$\Rightarrow x^2+7x-4x-28=874\times 3$
$\Rightarrow x^2+3x-2650=0$
$x=\frac{-3\pm \sqrt{(3{)}^2-4(-2650)}}{2}$
[by quadratic formula]
$=\frac{-3\pm \sqrt{9+10600}}{2}$
$=\frac{-3\pm \sqrt{10609}}{2}=\frac{-3\pm 103}{2}=\frac{100}{2},\frac{-106}{2}$
$=50,-53$
Here, x cannot be negative i.e., $x\ne -53$
Also, for x = - 53, n will be negative which is not possible.
Hence, the required value of x is 50.