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Question

The sum of the series 1·32+2·52+3·72+........upto 20 terms is


A

188090

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B

189080

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C

199080

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D

None of these

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Solution

The correct option is A

188090


Explanation for correct option

Let S=1·32+2·52+3·72+........

S=r=1nr(2r+1)2S=r=1n(4r2+1+4r)(r)S=r=1n(4r3+r+4r2)S=r=1n4r3+r=1n4r2+r=1nrS=4r=1nr3+4r=1nr2+r=1nrS=4n(n+1)22+4n(n+1)(2n+1)6+n(n+1)2S=4n2(n+1)24+23(2n3+2n2+n2+n)+n2+n2

as we know that r=1nr3=n(n+1)22,r=1nr2=n(n+1)(2n+1)6,r=1nr=n(n+1)2

Put n=20

S=(20)2(21)2+23(2×(20)3+2(20)2+(20)2+20)+(20)2+202S=400×441+23(16000+800+400+20)+400+202S=176400+11480+210S=188090

Hence, the correct option is optionA.


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