Mathematical Induction Basis Step Not Equal

Question

I am to prove the following statement using mathematical induction for all positive integers: $$ 1 - 2 + 2^2 - 2^3 +\dots+(-1)^n 2^n = \frac{2^n\cdot2^1\cdot(-1)^n+1}{3}. $$ However, for the basis step, I am getting $-2$ for the left side and $-1$ for the right side, which are obviously not equal. The assignment indicates that the statement is true though, so I am missing something or could this be a typo in the statement?

Answer 1

For $n=1$, you have to stop at the exponent $1$, so the left hand side is $$ 1+(-1)^1\cdot2^1 $$ and the right hand side is $$ \frac{2\cdot 2\cdot(-1)+1}{3}=\frac{-4+1}{3} $$ and both equal $-1$.

Whether you start at $1$ or at $0$ depends on your site's conventions or on the particular assignment. The formula also holds for $n=0$, when the left hand side is $1$ and the right hand side is $$ \frac{2^0\cdot2\cdot(-1)^0+1}{3}=\frac{2+1}{3}=1 $$