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 \(\frac{1-x^{2^{k+1}}}{1-x}\).

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Solution require uncommon idea
Cs translation
Send comment on task by email