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Question

Given that 1+i1+22i×1+32i1+42i×....×1+(2n1)2i1+(2n)2i=a+bic+di,showthat:
217×82257×....×(2n1)4+1(2n)4+1=a2+b2c2+d2.


Solution

1+i1+22i×1+321+42i×...×1+(2n1)2i1+(2n)2i=a+bic+di1+i1+22×1+32i1+42i×.....×1+(2n1)2i1+(2n)2i=a+bic+di1+i1+22i×1+32i1+42i×.....×1+(2n1)2i1+(2n)2i=a+bic+di.
||1+i1+22i.1+32i1+42i×...|1+(2n1)2i||1+(2n)2i|=|a+bi||c+di|1+11+16.1+811+256....(2n1)4+1(2n)4+1=a2+b2c2+d2217.82257...(2n1)4+1(2n)4+1=a2+b2c2+d2.

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