Find the sum to `n` terms of the series `1xx2xx3+2xx3xx4+3xx4xx5+dot`

Answer

N/a

Explanation

`1*2*3+2*3*4+3*4*5+….+` to n terms `T_(n)=`(nthe term of 1+2+3+..).(nth term of 1+3+4+...).(nth term of 3+4+5+…)
`{1+(n-1)*1}{2+(n-1)*1}{3+(n-1)*1}`
`=n(n+1)(n+2)`
`=n^(3)+3n^(2)+2n`
`implies S_(n)=sumn^(3)+3sumn^(2)+2sumn`
`=(1)/(4)n^(2)(n+1)^(2)+(3)/(6)n(n+1)(2n+1)+(2)/(2)n(n+1)`
`n(n+1)[(1)/(4)n(n+1)+(1)/(2)(2bn+1)+1]`
`=n(n+1)[(n(n+1)+2(2n+1)+4)/(4)]`
`=(1)/(4)n(n+1)(n(2)+n+4n+2+4)`
`=(1)/(4)n(n+1)(n^(2)+5n)+6)`
`(1)/(4)n(n+1)(n+2)(n+3)`