# Mathematical Inductions Problems

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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