Question

12
28.
Find the sum: 1.3 +2.4 +3.5+ ... to n terms. 2​

Answer 1

Let S denote the sum

`1\cdot3+2\cdot4+3\cdot5+\cdots+n\cdot(n+2)`

We can condense this to sigma notation:

`S=\displaystyle\sum_(k=1)^nk(k+2)`

Expand the summand:

`S=\displaystyle\sum_(k=1)^nk^2+2\sum_(k=1)^nk`

Recall the Faulhaber formulas,

`\displaystyle\sum_(k=1)^nk=\frac{n(n+1)}2`

`\displaystyle\sum_(k=1)^nk^2=\frac{n(n+1)(2n+1)}6`

So we have

`S=\frac{n(n+1)(2n+1)}6+n(n+1)=\boxed{\frac{n(n+1)(2n+7)}6}`