What is the $n$-th sequence for the following expression?

Question

We know that

$$\sum_{i=1}^n i = n(n+1)/2$$

Similarly, I want to find out an explicit expression for the following sum:

$$\sum_{i=1}^n(i+1)$$

Can anybody please help me?

Answer 1

For the sequence :$$\sum_{i=1}^n(i+1)$$

We can write,

$$\sum_{i=1}^n(i) + \sum_{i=1}^n(1) $$

From, $$\sum_{i=1}^n(i) = n(n+1)/2$$ , we can write

$$n(n+1)/2+n$$

Hence, $$\sum_{i=1}^n(i+1) = ( n^2+ 3n) /2$$

nth-term of the sequence : $$n+1$$