Question

Prove by induction:
$2+2^2+2^3+........+2^n=2(2^n-1)$

Answer 1

The statement to be proved is:

$P\left(n\right):2+2^2+2^3+...+2^n=2(2^n-1)$

Step 1: Prove that the statement is true for $n=1$

$P\left(1\right):2^1=2(2^1-1)$

$P\left(1\right):2=2$

Hence, the statement is true for $n=1$

Step 2: Assume that the statement is true for $n=k$

Let us assume that the below statement is true:

$P\left(k\right):2+2^2+...+2^k=2(2^k-1)$

Step 3: Prove that the statement is true for $n=k+1$

We need to prove that:

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

$LHS=2+2^2+...+2^k+2^{k+1}$

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

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

$=2({2.2}^k-1)$

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

$=RHS$

Therefore, $P\left(n\right)$ is true for all values of $n$ by principle of mathematical induction.