Basics of Induction

Prove the following using induction: 1. $\sum_{i=1}^{n}{i} = \frac{n(n+1)}{2}$. 2. $\sum_{i=1}^{n}{i}^2 = \frac{n(n+1)(2n+1}{6}$. 3. $\sum_{i=1}^{n}{i}^3 = {(\frac{n(n+1)}{2})}^2$. 4. $\sum_{i=1}^{n}{i}^4 = \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$. 5. $\frac{1}{1.2} + \frac{1}{2.3} + \frac{1}{3.4} + \dots + \frac{1}{n(n+1)} = \frac{n}{n+1}$ 6. $\sum_{i=0}^{n}{2i+1}^2 = \frac{(n+1)(2n+1)(2n+3)}{3}$. 7. For all $n \in \mathbb{N}$, $(n^3 - n)$ is divisible by $3$. 8. For all $n \in \mathbb{N}$, $(4^n + 15n - 1)$ is divisible by $9$.