# Principles of Mathematical Induction

Let n0 Є N and let P (n) be a statement for each natural number n ≥ n0.  Suppose that

• The statement P (n0) is true.
• For all n ≥ n0, P (K) is true => P (K + 1) is true.

Then P (n) is true for all n ≥ n0.

Cube: 1³ + 2³ + 3³ + … + n³ = $$\frac{{{n}^{2}}{{\left( n+1 \right)}^{2}}}{4}$$,   n ϵ N.

Sum of the ‘n’ numbers: 1 + 2 + 3 + … + n = $$\frac{n\left( n+1 \right)}{2}$$.

Square: 1² + 2² + 3² + … + n² =  $$\frac{n\left( n+1 \right)\left( 2n+1 \right)}{6}$$.