Solve the equation: -4+(-1)+......+x=437

(-4) + (-1) + 2 + 5 + ---- + x = 437.
Now,
-1 - (-4) = -1 + 4 = 3
2 - (-1) = 2 + 1 = 3
5 - 2 = 3
Thus, this forms an A.P. with a = -4, d = 3,l = x
Let their be n terms in this A.P.
Then,
S_n = $\frac { n } { 2 } [ 2 a + ( n - 1 ) d ]$
$\Rightarrow 437 = \frac { n } { 2 } [ 2 \times ( - 4 ) + ( n - 1 ) \times 3 ]$
$\Rightarrow$ 874 = n[-8 + 3n - 3]
$\Rightarrow$874 = n[3n - 11]
$\Rightarrow$874 = 3n² - 11n
$\Rightarrow$3n² - 11n - 874 = 0
$\Rightarrow$3n² - 57n + 46n - 874 = 0
$\Rightarrow$3n(n - 19) + 46(n - 19) = 0
$\Rightarrow$3n + 46 = 0 or n = 19
$\Rightarrow n = - \frac { 46 } { 3 }$ or n = 19
Numbers of terms cannot be negative or fraction.
$\Rightarrow$ n = 19
Now, S_n = $\frac { n } { 2 } [ a + l ]$
$\Rightarrow 437 = \frac { 19 } { 2 } [ - 4 + x ]$
$\Rightarrow - 4 + x = \frac { 437 \times 2 } { 19 }$
$\Rightarrow - 4 + x = 46$
$\Rightarrow x = 50$