Find the value of the expression `sum_(i=1)^(n)sum_(j=1)^(i)sum_(k=1)^(j)1`.

Options

A. `sumn`
B. `sumn^(2)`
C. `sumn^(3)`
D. none of these

Answer

D

Explanation

We have,
`underset(i=1)overset(n)sumunderset(j=1)overset(i)sumunderset(k=1)overset(j)sum1=underset(i=1)overset(n)sumunderset(j=1)overset(i)sumj`
`=underset(i=1)overset(n)sum(1+2+. . . .i)`
`underset(i=1)overset(n)sum(i(i+1))/(2)=(1)/(2){underset(i=1)overset(n)sumi^(2)+underset(i=1)overset(n)sumi}`
`=(1)/(2){(n(n+1)(2n+1))/(6)+(n(+1))/(2)}`
`=(n(n+1))/(4){(2n+1)/(3)+1}=(n(n+1)(n+2))/(6)`