## Algebra and Trigonometry 10th Edition

The formula was proved for $n=1$ The formula is correct if $n$ is changed by $n+1$
Let's prove the formula for $n=1$: $2^1-1=2-1=1$ It is correct! Now, suppose that the formula is correct, that is: $1+2+2^2+2^3+...+2^{n-1}=2^n-1$ Now, let's prove the formula for $n+1$: $1+2+2^2+2^3+...+2^{n-1}+2^{(n+1)-1}=(1+2+2^2+2^3+...+2^{n-1})+2^n=2^n-1+2^n=2(2^n)-1=2^1(2^n)-1=2^{n+1}-1$ That is exactly the given formula if $n$ is changed by $n+1$