What would be the sum of following?

Question

How to calculate the sum of following:

$$n + 2(n-1) + 3(n-2) + \cdots + n$$

The sum neither seems to fit standard definition of AP nor GP.

Like $$1 + 2 + \cdots +n = n(n+1)/2$$

Answer 1

Observe that this sum is, more explicitly,

$$\sum_{k=1}^n k(n-k+1)$$ Manipulate:

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

$$\sum_{k=1}^n (-k^2 + kn + k)$$

$$\sum_{k=1}^n (-k^2) + \sum_{k=1}^n kn + \sum_{k=1}^n k$$

$$-\sum_{k=1}^n k^2 + n \sum_{k=1}^n k + \sum_{k=1}^n k$$ $$-\sum_{k=1}^n k^2 + (n +1)\sum_{k=1}^n k$$

$$-\frac{n (n + 1) (2 n + 1)}6 + (n+1) \frac{n(n+1)}2$$

$$\frac{n (n + 1) (n + 2)}6$$