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

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Solution

Common difference $= a_{2} - a_{1} = (- 1) - (- 4) = 3$
First term $= (- 4)$
$(S u m)_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} [2 (- 4) + (n - 1) 3]$
$S_{n} = \frac{n}{2} (- 8 + 3 n - 3) = \frac{n (3 n - 11)}{2} 437 = \frac{n (3 n - 11)}{2} 3 n^{2} - 11 n - 874 = 0 n = \frac{+ 11 \pm \sqrt{(121) + 12 \times 874}}{6} = \frac{11 \pm \sqrt{10609}}{6} n = \frac{11 + 103}{6} = \frac{114}{6} = 19 n = 19 x = a_{1} + (n - 1) d = - 4 + (19 - 1) 3 x = - 4 + 18 \times 3 x = - 4 + 54 x = 50$

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