Term adjustment
Task number: 3867
Simplify as much as possible \((1+x)(1+x^2)(1+x^4)\ldots(1+x^{2^k})\).
Solution
Substitution for small values of \(k\) yields:
- \((1+x)(1+x^2)=1+x+x^2+x^3\)
- \((1+x)(1+x^2)(1+x^4)=(1+x+x^2+x^3)(1+x^4)=1+x+x^2+…+x^7\).
By induction we show that \((1+x)(1+x^2)(1+x^4)\ldots(1+x^{2^k})=1+x+x^2+…+x^{2^{k+1}-1}\).
\((1+x)(1+x^2)(1+x^4)\ldots(1+x^{2^k})= (1+x)(1+x^2)(1+x^4)\ldots(1+x^{2^{k-1}})(1+x^{2^k})=\\ (1+x+x^2+…+x^{2^k-1})(1+x^{2^k})= (1+x+x^2+…+x^{2^k-1})+x^{2^k}(1+x+x^2+…+x^{2^k-1})=\\ 1+x+x^2+…+x^{2^{k+1}-1}\)
Finally, we use the formula for the sum of the geometric series to get the result.
Answer
The term can be simplified to \(\frac{1-x^{2^{k+1}}}{1-x}\).
