<s>[INST] Prove equation with induction Prove by induction that for every natural number $n\in\mathbb{N}$ and every real number $x\in\mathbb{R}$:
$$(1+x)(1+x^2)(1+x^4)\cdot\,\dots\,\cdot(1+x^{2^{n-1}}) = 1 +x +x^2+\dots+(x^{2^n-1})$$
I proved for $n=1$ but i don't know how to continue with $(n+1)$
THANKS
 [/INST] I think you made some mistake in the formula. We prove 
$$(1+x)(1+x^2)\cdots (1+x^{2^{n-1}}) = 1+x+x^2+x^3+\cdots + x^{2^n-1}.$$
For $n=1$ we get $x+1=x+1$ which holds.
Assume the statement holds for $n$. We prove it holds for $n+1$. So we have
$$(1+x)(1+x^2)\cdots(1+x^{2^{n-1}})(1+x^{2^{n}}) = (1+x+\cdots +x^{2^n-1})(1+x^{2^{n}}),$$
by our induction assumption. But
$$(1+x+\cdots +x^{2^n-1})(1+x^{2^{n}})=1+x+\cdots +x^{2^n-1} + x^{2^n}+x^{2^n+1} \cdots + x^{2^n+2^{n-1}}=1+x+\cdots+x^{2^{n+1}-1},$$
proving the statement.
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