Prove the following by the principle of mathematical induction:`\ 1+2+2^2.......+2^n=2^(n+1)-1` for all `n in Ndot`

Explanation

`{Letp}(n): 1+2+2^{2}+ldots+2^{n}=2^{n+1}-1 forall n in N`

Step I: For `n=1`,

`L H S=1+2^{1}=3`

`R H S=2^{1+1}-1=2^{2}-1=4-1=3`

`A s, L H S=R H S`

So, it is true for `n=1`.

Step Il: For `{n}=k`,

Let `p(k): 1+2+2^{2}+ldots+2^{k}=2^{k+1}-1` be true `forall k in N`

...