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Easy

Question

The sum to 50 terms of the series $1 plus 2 open parentheses 1 plus fraction numerator 1 over denominator 50 end fraction close parentheses plus 3 open parentheses 1 plus fraction numerator 1 over denominator 50 end fraction close parentheses to the power of 2 end exponent plus horizontal ellipsis$ is given by

2500
2550
2450
None of these

The correct answer is: 2500

Let $1 plus 1 divided by 50 equals x$ . Let $S$ be the sum of 50 terms of the given series. Then,
$S equals 1 plus 2 x plus 3 x to the power of 2 end exponent plus 4 x to the power of 3 end exponent plus horizontal ellipsis plus 49 x to the power of 48 end exponent plus 50 x to the power of 49 end exponent$ (1)
$x S equals x plus 2 x to the power of 2 end exponent plus 3 x to the power of 3 end exponent plus horizontal ellipsis plus 49 x to the power of 49 end exponent plus 50 x to the power of 50 end exponent$ (2)
$_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _$
$open parentheses 1 minus x close parentheses S equals 1 plus x plus x to the power of 2 end exponent plus x to the power of 3 end exponent plus horizontal ellipsis plus x to the power of 49 end exponent minus 50 x to the power of 50 end exponent$ [Subtracting (2) from (1)]
$rightwards double arrow S open parentheses 1 minus x close parentheses equals fraction numerator 1 minus x to the power of 50 end exponent over denominator 1 minus x end fraction minus 50 x to the power of 50 end exponent$
$rightwards double arrow S left parenthesis negative 1 divided by 50 right parenthesis equals negative 50 left parenthesis 1 minus x to the power of 50 end exponent right parenthesis minus 50 x to the power of n end exponent$
$rightwards double arrow fraction numerator 1 over denominator 50 end fraction S equals 50$
$rightwards double arrow S equals 2500$

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