Question

find the value of x:
x + (-2x) + (-5x) +.... + (-71x) = 1750​

Answer 1

$Given \: x + (-2x) + (-5x) +\ldots + (-71x) = 1750$

$Here, First \:term ( a = a_{1}) = x$

$a_{2} - a_{1} = -2x - x = -3x \: --(1)$

$and \: a_{3} - a_{2} = -5x -(-2x) = -3x \: --(2)$

$a_{2} - a_{1} = a_{3} - a_{2} = -3x$

$\therefore Given \: sequence \: is \: in \: A.P$

$Common \: difference (d) = -3x$

$n^{th} \:term = -71x$

$\implies a + (n-1)d = -71x$

$\implies x + (n-1)(-3x) = -71x$

$\implies (n-1)(-3x) = -71x - x$

$\implies (n-1)(-3x) = -72x$

$\implies n-1 = \frac{-72x}{-3x}$

$\implies n-1 = 24$

$\implies n = 24 + 1$

$\implies n = 25$

/* We know that */

$\boxed{ \pink{ Sum \: of \:n \: terms (S_{n}) = \frac{n}{2} ( a + a_{n}) }}$

$Here , a = x , a_{n} = -71x , n = 25$

$S_{25} = 1750 \:( given )$

$\implies \frac{25}{2} [ x - 71x ] = 1750$

$\implies \frac{25}{2} [ - 70x ] = 1750$

$\implies 25 \times ( -35 x ) = 1750$

$\implies x = \frac{1750}{ 25 \times (-35)}$

$\implies x = -2$

Therefore.,

$\red{ Value \: of \: x } \green { = -2 }$

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