Some typical calculation problems often appear in entry to intermediate level competitions. Some examples are shown below:

(4131) Compute: $1\times 2\times 3 + 2\times 3\times 4 + \cdots + 18\times 19\times 20$.

(3269) Let $$f(r) = \displaystyle\sum_{j=2}^{2008}\frac{1}{j^r} = \frac{1}{2^r}+\frac{1}{3^r}+\cdots+\frac{1}{2016^r}$$Find $$\sum_{k=2}^{\infty}f(k)$$

(3240) Evaluate $\sqrt{5+\sqrt{5^2+\sqrt{5^4+\sqrt{5^8+...}}}}$

This attachment includes $8$ such typical problems (including some trigonometric and combinatorics expressions) and their complete solutions. More thorough discussions about various techniques in this subject can be found in the book  Power Calculation .