<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 can't resist to show non purely inductive proof. Denote
$$
P=(1+x)(1+x^2)\cdot\ldots\cdot(1+x^{2^{n-1}})
$$
Note that
$$
\begin{align}
(1-x)P&=(1-x)(1+x)(1+x^2)(1+x^4)\cdot\ldots\cdot(1+x^{2^{n-1}})\\
&=(1-x^2)(1+x^2)(1+x^4)\cdot\ldots\cdot(1+x^{2^{n-1}})\\
&=(1-x^4)(1+x^4)\cdot\ldots\cdot(1+x^{2^{n-1}})\\
&=\ldots\\
&=(1-x^{2^{n-1}})(1+x^{2^{n-1}})\\
&=1-x^{2^n}
\end{align}
$$
Hence
$$
P=\frac{1-x^{2^n}}{1-x}=1+x+x^2+\ldots+x^{2^n-1}
$$
</s>