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I have posted a link there to this thread so the OP can view my work.Please help to prove by induction?

Please explain me how to prove this by induction. Thanks a lot!

Use the fact that \(\displaystyle (x+y)(x-y)=x^2-y^2\) to prove by induction that:

\(\displaystyle \prod_{k=0}^n\left(1+x^{2^k} \right)=\frac{1-x^{2^{n+1}}}{1-x}\)

for any \(\displaystyle n\in\mathbb{N}\) and any \(\displaystyle x\in\mathbb{Q}\) with \(\displaystyle x\ne1\).