Find the sum of the series: `1. n+2.(n-1)+3.(n-2)++(n-1). 2+n .1.`

Answer

N/a

Explanation

`therefore" "T_(r)={"rth term of 1, 2, 3, ..."}xx{"rth term of n," (n-1), (n-2)...}`
`=rcdot{n+(r-1)(-1)}`
`=r(n+1-r)`
`=(n+1)cdotr-r^(2)`
`therefore" "S_(n)=overset(n)underset(r=1)SigmaTr=overset(n)underset(r=1)Sigma[(n+1)r-r^(2)]`
`=(n+1)overset(n)underset(r=1)Sigmar-overset(n)underset(r=1)Sigmar^(2)`
`=(n+1)cdot(n)/(2)(n+1)-(n)/(6)(n+1)(2n+1)`
`=n(n+1)[(n+1)/(2)-(2n+1)/(6)]`
`=n(n+1)[(3n+3-2n-1)/(6)]`
`=(1)/(6)n(n+1)(n+2).`