Chapter 12.5, Problem 14E

To determine

To Prove: $1+2+2^2+\mathrm{.....}+2^{n-1}=2^n-1$ is True for all natural numbers $n$ .

Explanation of Solution

Given information:

Use Mathematical induction to prove the above mentioned formula.

Concept used:

1.By using Mathematical induction technique we need to prove that this formula holds True for all natural numbers. The first step is to prove True for $n=1$ .

2.Second step by using induction hypothesis of mathematical induction we assume for $n=k$ given formula is True.

3.Third step is we need to prove that above formula is True for $n=k+1$ by using formula is True for $n=k$ .

Proof:

First, we need to prove the above formula is True for $n=1$ .

Given formula is

  $1+2+2^2+\mathrm{.....}+2^{n-1}=2^n-1$

Put $n=1$

We get

Left hand side = 1

Right hand side = $2^1-1=2-1=1$

So Left hand side = Right hand side

It True for $n=1$ .

Let us assume it is True for $n=k$

So

  $1+2+2^2+\mathrm{.....}+2^{k-1}=2^k-1$ is True

We need to prove it is True for $n=k+1$

Left hand side = $1+2+2^2+\mathrm{.....}+2^{k-1}+2^{k+1-1}$

  $\Rightarrow$ Left hand side = $2^k-1+2^k$

  $\Rightarrow$ Left hand side = $2\times 2^k-1$

  $\Rightarrow$ Left hand side = $2^{k+1}-1$

Right hand side = $2^{k+1}-1$

Left hand side = Right hand side

And hence the proof by Mathematical induction.