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Question

The sum of the product of the integers 1,2,3,...., n taken two at a time is



A

n(n+1)24(3n2n2)

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B

n2(n+1)22(n2+1)

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C

n(n+1)18(n2n2)

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D

n(n+1)(n+2)(n+3)24

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Solution

The correct option is A

n(n+1)24(3n2n2)


(a+b)2=(a2+b2)+2(ab)=a2+2ab(a+b+c)2=(a2+b2+c2)+2(ab+bc+cd)=a2+2abIn general,(a1+a2++an)2=(a21+a22++a2n)+2(a1a2+a1a3++an1an)=a21+2a1a2(1+2+3++n)2=(12+22+32++n2)+2(sum of products of n integers taken two at a time)
i.e. (n)2=n2+2S; where S is the required sum
S=12[(n)2(n2)]=12[{n(n+1)2}2{n(n+1)(2n+1)6}]=n(n+1)24(3n2n2)


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