Question

$1+2+2^2+...+2^n=2^{n+1}-1$ for all $nϵN$.

Answer 1

Let P(n) : $1+2+2^2+....+2^n=2^{n+1}-1$ for all natural numbers n.

We observe that P(0) is true.

$P\left(0\right):1=2^{0+1}-1$

$1=2^1-1$

$1=2-1$

$1=1$, which is true.

Now, assume that P(n) is true for n = k.

So, P(k) : $1+2+2^{2+}+.....+2^k=2^{k+1}-1$ is true.

Now, We shall prove P(k + 1) is true.

$P(k+1):1+2+2^2+....+2^k+2^{k+1}$

$=2^{k+1}-1+2^{k+1}$

$=2^{k+2}-1$

$=2^{(k+1)+1}-1$

So, P(k+1) is true whenever P(k) is true.

Hence, P(n) is true.