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## 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 $9$.

9. Prove that if $x \geq -1$, then for any integer $n \geq 0$, $(1 + x) ^ n > n x$.

Source: folklore

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