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Algebra Level 3

Which of the following is equivalent to

( x + 1 ) ( x 2 + 1 ) ( x 4 + 1 ) ( x 8 + 1 ) . . . ( x 65536 + 1 ) (x+1)(x^{2} + 1)(x^{4} + 1)(x^{8} + 1)...(x^{65536} + 1) ?

x 131072 + 1 x + 1 \frac{x^{131072} + 1}{x + 1} x 131072 − 1 x^{131072} - 1 x 131072 + x 65536 + . . . + x 2 + 1 x^{131072} + x^{65536} + ... + x^{2} + 1 x 131072 − 1 x − 1 \frac{x^{131072} - 1}{x - 1}

Sanchayapol Lewgasamsarn by Sanchayapol Lewgasamsarn , 20, Thailand

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1 solution

Let the whole thing be named S.

( x − 1 ) S = ( x 131072 − 1 ) (x - 1)S = (x^{131072} - 1) , because 65536 = 2 16 65536 = 2^{16}

So, S = x 131072 − 1 x − 1 S = \boxed{\frac{x^{131072} - 1}{x-1}}

15 Helpful 0 Interesting 1 Brilliant 0 Confused

a beautiful solution!!!!!

Mayank Holmes - 6 years, 11 months ago

Gorgeous solution.

Ryan Tamburrino - 6 years, 10 months ago

WHA~ That's it? Whoa. Great!

Joeie Christian Santana - 6 years, 8 months ago

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