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Question

i=1nj=11k=1j1 is equal to


A

n(n+1)2

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B

n(n+1)(n+2)6

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C

n(n+1)22

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D

None of these

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Solution

The correct option is B

n(n+1)(n+2)6


Explanation for the correct answer:

Simplifying the given expression:

i=1nj=11k=1j1i=1nj=11jk=1j1=1i=1n(1+2+3+.....+i)[Considerj=1+2+3+...+i]i=1ni(i+1)2sumofnnaturalnumbersisn(n+1)2i=1ni2+i2n(n+1)(2n+1)12+n(n+1)4sumofsquaresofnnaturalnumbers=n(n+1)(2n+1)2n(n+1)42n+13+1n(n+1)(2n+4)4×3n(n+1)(n+2)6

Thus, i=1nj=11k=1j1=n(n+1)(n+2)6

Hence, option (B) is correct.


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