Question

# Assertion :12(1)(3)+22(3)(5)+.....+n2(2n−1)(2n+1)=n(n+1)2(2n+1) because Reason: 1(1)(3)+1(3)(5)+.....+1(2n−1)(2n+1)=12n+1

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Assertion is incorrect but Reason is correct
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Solution

## The correct option is C Assertion is correct but Reason is incorrectLet tr=r2(2r−1)(2r+1)⇒4tr=4r2−1+1(2r−1)(2r+1)=1+1(2r−1)(2r+1)=1+12(12r−1−12r+1)4∑nr=1tr=n+12∑nr=1(12r−1−12r+1)=n+12(1−12n+1)=n+n2n+1=2n(n+1)2n+1⇒∑nr=1tr=n(n+1)2(2n+1)∴ Statement -1 is trueBut statement-2 is false since⇒∑nr=11(2r−1)(2r+1)=12∑nr=1(12r−1−12r+1)=12(1−12n+1)=n2n+1

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