How to solve this summation derived from an algorithm?

Question

I have the following expression:

$$\sum_{i=1}^{n}\sum_{j=1}^{i^2}\sum_{k=1}^{n/2}c=$$

Can someone explain in detail how can I solve this one?

Answer 1

HINT

Note that for constant terms

$$\sum_{k=1}^{n/2}c=c\sum_{k=1}^{n/2}1=\frac{cn}2$$

$$\sum_{j=1}^{i^2}\sum_{k=1}^{n/2}c=\sum_{j=1}^{i^2}\frac{cn}2=\frac{cn}2\sum_{j=1}^{i^2}1=\frac{cni^2}{2}$$

and then

$$\sum_{i=1}^{n}\sum_{j=1}^{i^2}\sum_{k=1}^{n/2}c=\frac{cn}2\sum_{i=1}^{n}i^2$$

then use the well known formula for $\sum_{i=1}^{n}i^2$.