Principle of Ma...

If $$P(n)$$ is statement such that $$P(3)$$ is true. Assuming P(k) is true $$\Rightarrow$$ $$P(k+1)$$ is true for all $$k$$ $$\geq$$ $$2$$, then $$ P(n)$$ is true.

$$\displaystyle \frac{1}{2} + \frac{1}{4}+ \frac{1}{8} + ......... + \frac{1}{2^n} = 1 - \frac{1}{2^n}$$ is true for

$$1.3 + 3.5 + 5.7 + ........... + (2n -1) (2n + 1) = \displaystyle \frac{n (4n^2 + 6n -1)}{3}$$ is true for

$$\forall n\in N, 2 \cdot 4^{2n + 1} + 3^{3n + 1}$$ is divisible by

$$\forall n\in N, 3^{3n} - 26^{n}$$ is divisible by

$$7^{2n} + 3^{n - 1} \cdot 2^{3n - 3}$$ is divisible by

$$\forall n\in N; 10^{2n - 1}+1$$ is divisible by

If $$x^n - 1$$ is divisible by $$x - k$$, then the least positive integral value of $$k$$ is

$$\forall n\in N, n^4$$ is less than

If  $$n$$ $$ \in$$ N, then $$x^{2n - 1} + y^{2n - 1}$$ is divisible by