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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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