Sum of the following series to `n` term: `2+4+7+11+16+ `

Explanation

The sequence of first consctive differences is `2,3,4,5,"…"`. Clearly, it is an AP. Then, nth term of the given series be
`T_(n)=a(n-1)(n-2)+b(n-1)+c " " "....(i)"`
Putting `n=1,2,3,` we get
`2=c implies 4=b+c implies 7=2a+2b+c`
After solving, we get `a=(1)/(2),b=2,c=2`
Putting the values of `a,b,c` in Eq. (i), we get
`T_(n)=(1)/(2)(n-1)(n-2)+2(n-1)+2=(1)/(2)(n^(2)+n+2)`
Hence, sum of series `S_(n)=sum T_(n)=(1)/(2)(sumn^(2)+sumn+2sum1)`
`=(1)/(2)((n(n+1)(2n+1))/(6)+(n(n+1))/(2)+2n)`
`=(1)/(6)n(n^(2)+3n+8)`