I need help proving the base case for a mathematical induction proof

Question

I know how mathematical induction works and the generic algorithm of proving a statement by the Principle of Mathematical Induction, but I'm having trouble proving the base case for a particular problem in my textbook.

For every $n\in\Bbb{Z}^+$ where $x\neq1$
$$(1+x)(1+x^2)(1+x^4)...(1+x^{2n})=\frac{1-x^{2n+1}}{1-x}$$

My attempt for base case $n=1$:
$LHS: (1+x)(1+x^2)=1+x+x^2+x^3$
$RHS: \frac{1-x^3}{1-x}=\frac{(1-x)(1+x+x^2)}{1-x}=1+x+x^2$

Which is obviously not true for all $x\neq1$

What am I doing wrong?

Answer 1

The claim is wrong. The correct statement is $$(1+x^{2^0})(1+x^{2^1})(1+x^{2^2})...(1+x^{2^n})=\frac{1-x^{2^{n+1}}}{1-x}$$

Answer 2

The correct formula you should be proving is:

$$(1+x)(1+x^2)(1+x^4)...(1+x^{2^n})=\frac{1-x^{2^{n+1}}}{1-x} $$

You either copied the problem wrong, or there is a mistake in the problem.