4. Use mathematical induction to prove\[ 2 - 2 \times 9 + 2 \times 9^2 - \cdots + 2 \times (-9)^n = \frac{1 - (-9)^{n+1}}{5} \]whenever ? is a nonnegative integer. (10%)

(b) The binary relation can be represented graphically. Represent the elements in Tby points and use arrows to represent the ordered pairs in R. Please show that the above-mentioned directed graph has no cycles if deleting the self-loops from the poset (T, R). (5%)