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Maths-

General

Easy

Question

The sum of the series $1 times 3 to the power of 2 end exponent plus 2 times 5 to the power of 2 end exponent plus 3 times 7 to the power of 2 end exponent plus midline horizontal ellipsis$ up to 20 terms is

188090
189080
199080
None of these

The correct answer is: 188090

$text Here end text T subscript n end subscript text of the AP end text 12 comma 3 comma midline horizontal ellipsis equals n$
And $T subscript n end subscript$ of the AP 3, 5, 7, … = 2n+1
Therefore,
$T subscript n end subscript text of given series end text equals n left parenthesis 2 n plus 1 right parenthesis to the power of 2 end exponent equals 4 n to the power of 3 end exponent plus 4 n to the power of 2 end exponent plus n$
$text Hence, end text S equals stretchy sum from 1 to 20 of T subscript n end subscript equals 4 stretchy sum from 1 to 20 of n to the power of 3 end exponent plus 4 stretchy sum from 1 to 20 of n to the power of 2 end exponent plus stretchy sum from 1 to 20 of n$
$equals 4 times fraction numerator 1 over denominator 4 end fraction 20 to the power of 2 end exponent times 21 to the power of 2 end exponent plus 4 times fraction numerator 1 over denominator 6 end fraction 20 times 21 times 41 plus fraction numerator 1 over denominator 2 end fraction 20 times 21 equals 188090$

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