The power of 2

Algebra Level 3

( 1 − x 32 ) ( 1 1 − x + 1 x + 1 + 2 x 2 + 1 + 4 1 + x 4 + 8 1 + x 8 + 16 1 + x 16 ) = ? \left(1 - x^{32}\right)\left(\frac{1}{1 - x} + \frac{1}{x + 1} + \frac{2}{x^2 + 1} + \frac{4}{1 + x^4} + \frac{8}{1 + x^8} + \frac{16}{1 + x^{16}}\right) = ?

This is part of the series: It's easy, believe me!


The answer is 32.

Thành Đạt Lê by Thành Đạt Lê , 17, Singapore

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1 solution

Chew-Seong Cheong
Aug 15, 2017

X = ( 1 − x 32 ) ( 1 1 − x + 1 1 + x + 2 1 + x 2 + 4 1 + x 4 + 8 1 + x 8 + 16 1 + x 16 ) = ( 1 − x 32 ) ( 1 + x + 1 − x ( 1 − x ) ( 1 + x ) + 2 1 + x 2 + 4 1 + x 4 + 8 1 + x 8 + 16 1 + x 16 ) = ( 1 − x 32 ) ( 2 1 − x 2 + 2 1 + x 2 + 4 1 + x 4 + 8 1 + x 8 + 16 1 + x 16 ) = ( 1 − x 32 ) ( 4 1 − x 4 + 4 1 + x 4 + 8 1 + x 8 + 16 1 + x 16 ) = ( 1 − x 32 ) ( 8 1 − x 8 + 8 1 + x 8 + 16 1 + x 16 ) = ( 1 − x 32 ) ( 16 1 − x 16 + 16 1 + x 16 ) = ( 1 − x 32 ) ( 32 1 − x 32 ) = 32 \begin{aligned} X & = \left(1-x^{32}\right)\left(\frac 1{1-x} + \frac 1{1+x} + \frac 2{1+x^2} + \frac 4{1+x^4} + \frac 8{1+x^8} + \frac {16}{1+x^{16}} \right) \\ & = \left(1-x^{32}\right)\left(\frac {1+x+1-x}{(1-x)(1+x)} + \frac 2{1+x^2} + \frac 4{1+x^4} + \frac 8{1+x^8} + \frac {16}{1+x^{16}} \right) \\ & = \left(1-x^{32}\right)\left(\frac 2{1-x^2} + \frac 2{1+x^2} + \frac 4{1+x^4} + \frac 8{1+x^8} + \frac {16}{1+x^{16}} \right) \\ & = \left(1-x^{32}\right)\left(\frac 4{1-x^4} + \frac 4{1+x^4} + \frac 8{1+x^8} + \frac {16}{1+x^{16}} \right) \\ & = \left(1-x^{32}\right)\left(\frac 8{1-x^8} + \frac 8{1+x^8} + \frac {16}{1+x^{16}} \right) \\ & = \left(1-x^{32}\right)\left(\frac {16}{1-x^{16}} + \frac {16}{1+x^{16}} \right) \\ & = \left(1-x^{32}\right)\left(\frac {32}{1-x^{32}} \right) \\ & = \boxed{32} \end{aligned}

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A lovely problem if you don't know the answer is a particular number. But, in the context of being asked for an answer we can type into the form, there is a cheat: let x = 0.

If we get a particular answer whatever the value of x, then any value will reveal that answer. 0 is the easiest to calculate. Of course, x can't be 1.

Samir Betmouni - 3 years, 7 months ago

x can't be 1

I Gede Arya Raditya Parameswara - 3 years, 3 months ago

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