Question

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

Answer 1

Step-by-step explanation:

-4+(-1) +2+x+x=437

= -3+2x=437

= 2x=437+3

=2x=440

=x=440/2

=220

Answer 2

Given eqn. is

-4 + (-1) + 2 + ... + x = 437 ____ (i)

Here , -4 - 1 + 2 + ... + x forms an AP with first term = -4 , common difference = 3 ,

${a}_{n}$ = I + x

$\therefore$ nth term of an AP , ${a}_{n} = l = a + (n-1)d \\\\\ \implies x = -4 + (n-1)3 \\\\\ \implies \frac {x+4}{3} = n - 1 \implies n = \frac {x+7}{3} \\\\\ \therefore Sum~ of ~ an ~ AP ~ , {S}_{n} = \frac {n}{2} [ 2a + (n-1)d] \\\\\ {S}_{n} = \frac {x+7}{2 \times 3} [ 2(-4) + (\frac{x+4}{3}).3] \\\\\ = \frac {x+7}{2 \times 3} ( -8+x+4) = \frac{(x+7)(x-4)}{2 \times 3}$

From eqn. (i) ,

${S}_{n} = 437 \\\\\ \implies \frac {(x+7)(x-4)}{2 \times 3 } = 437 \\\\\ \implies x² + 7x - 4x - 28 = 874 \times 3 \\\\\ \implies x² + 3x - 2650 = 0 \\\\\ x = \frac { -3±{\sqrt{(3)²-4(-2650)}}}{2}$

By quadratic formula ,

$= \frac {-3 ± {\sqrt {9 + 10600}}}{2} \\\\\ = \frac{-3±{\sqrt {1069}}}{2} = \frac {-3±103}{2} = \frac {100}{2} , \frac {-106}{2} \\\\\ = 50 , -53$

Here , x cannot be negative i.e. , $x ≠ -53$

Also , for x = -53 , n will be negative which is not possible .

Hence , the required value of x is 50 .