Prove the Following using Principle of Mathematical induction
1. $2 + 4 + 6 + ... + 2n = n^2 + n$ for all natural numbers n.
2. $1 + 3 + 3^2+ ... + 3^n= \frac {1}{2}(3^{n+1} – 1) for all natural numbers n.
3. $2n + 1 < 2^n$ , for all natual numbers n = 3.
4. n(n+1)(n+5) is divisible by 6 for all natural numbers n.
5. $1.2.3+ 2.3.4 + 3.4.5 +.....+ n(n+1)(n+2) = \frac {n(n+1)(n+2)(n+3)}{4}$
6. $sin \theta + sin 3\theta+ .....+ sin(2n-1) \theta = \frac {sin^2 n \theta}{sin \theta} $ for all natural numbers n.
7. $ 10^{2n-1} + 1$ is divisible by 11 for all natural numbers n.
8. $ 1 + 2 + ... + n = \frac {n(n+1)}{2}$ for all natural numbers n.
9. $n^2 - 3n + 4$ is even and it is true for all positive integers.
10. $\frac {1}{1 \times 2} + \frac {1}{2 \times 3 } + \frac {1}{3 \times 4 } +... + \frac {1}{n(n+1) } = \frac {n}{n+1}$ for all natural numbers n.
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