True or False (Problem from 'What is mathematics')

Algebra Level 3

( 1 + q ) ( 1 + q 2 ) ( 1 + q 4 ) . . . ( 1 + q 2 n ) = 1 − 2 q 2 n + 1 1 − q (1+q)(1+q^2)(1+q^4)...(1+q^{2^n})= \frac {1-2q^{2^{n+1}}}{1-q}

Is the above true or false?

true false

Jyothiraditya Chaitanya by Jyothiraditya Chaitanya , 16, India

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1 solution

Atomsky Jahid
Jun 17, 2019

This process is like an avalanche. The most crucial part is ( 1 + q ) = 1 − q 2 1 − q (1+q) = \frac{1-q^2}{1-q} After that, join the terms one by one. ( 1 − q 2 ) ( 1 + q 2 ) = ( 1 − q 4 ) (1-q^2)(1+q^2) = (1-q^4) ( 1 − q 4 ) ( 1 + q 4 ) = ( 1 − q 8 ) (1-q^4)(1+q^4) = (1-q^8) ⋮ \vdots ( 1 − q 2 n ) ( 1 + q 2 n ) = ( 1 − q 2 n + 1 ) (1-q^{2^n})(1+q^{2^n}) = (1-q^{2^{n+1}}) Use all these facts to get the identity ( 1 + q ) ( 1 + q 2 ) ( 1 + q 4 ) ⋯ ( 1 + q 2 n ) = 1 − q 2 n + 1 1 − q (1+q) (1+q^2) (1+q^4) \cdots (1+q^{2^n}) = \frac{1-q^{2^{n+1}}}{1-q}

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Or you could have used induction

Jyothiraditya Chaitanya - 1 year, 11 months ago

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Yep! But, this is more intuitive. Also, induction starts with a known identity. If you didn't know the result in the first place, how would you deduce the identity?

Atomsky Jahid - 1 year, 11 months ago

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