Solve the equation -4 + (-1) + 2 + ldots + x | vedclass

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Question

(B) The given series is $-4, -1, 2, \ldots, x$,which forms an Arithmetic Progression $(AP)$.Here,the first term $a = -4$ and the common difference $d

Vedclass · Accepted Answer

$50$

Vedclass · Answer

(B) The given series is $-4, -1, 2, \ldots, x$,which forms an Arithmetic Progression $(AP)$.Here,the first term $a = -4$ and the common difference $d = (-1) - (-4) = 3$.Let the number of terms be $n$. The $n^{th}$ term is $x = a + (n - 1)d$.$x = -4 + (n - 1)3 \Rightarrow x = -4 + 3n - 3 \Rightarrow x = 3n - 7 \Rightarrow n = \frac{x + 7}{3}$.The sum of the $AP$ is given by $S_n = \frac{n}{2}(a + l)$,where $l = x$ is the last term.$437 = \frac{n}{2}(-4 + x)$.Substituting $n = \frac{x + 7}{3}$ into the equation:$437 = \frac{x + 7}{2 	imes 3}(-4 + x) = \frac{(x + 7)(x - 4)}{6}$.$437 	imes 6 = x^2 + 3x - 28$.$2622 = x^2 + 3x - 28 \Rightarrow x^2 + 3x - 2650 = 0$.Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:$x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-2650)}}{2(1)} = \frac{-3 \pm \sqrt{9 + 10600}}{2} = \frac{-3 \pm \sqrt{10609}}{2}$.$x = \frac{-3 \pm 103}{2}$.Since $x$ must be positive for the series to sum to $437$,we take $x = \frac{100}{2} = 50$.

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

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