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Find the sum sum(r=1)^n r(r+1)(r+2)(r+3)...

Find the sum `sum_(r=1)^n r(r+1)(r+2)(r+3)`

Text Solution

Verified by Experts

`T_(r)=r(r+1)(r+2)(r+3)`
`=1/5r(r+1)(r+2)(r+3)(r+4)-(r-1)`
`=1/5(r(r+1)(r+2)(r+3)(r+4)-(r-1)r(r+1)(r+2)(r+3)`
`=1/5(V(r+1)-V(r ))`,
where `V(r )=(r-1)r(r+1)(r+2)(r+3)`
`therefore` Required Sum, `sum_(r=1)^(n)T_(r)=1/5(V(n)-V(l))`
`=1/5(n(n+1)(n+2)(n+3)(n+4)-0)`
`=1/5n(n+1)(n+2)(n+3)(n+4)`
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