A curious identity : 2

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Let w = e π i / 2015 \displaystyle w=e^{\pi{i}/2015}

Then evaluate the product ∏ k = 1 2014 ( w k + w − k ) \displaystyle \prod_{k=1}^{2014}(w^k+w^{-k})

This problem is inspired from Here


The answer is -1.

Pradeep Maurya by Pradeep Maurya , 32, India

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1 solution

Otto Bretscher
May 6, 2015

I guess we think alike: I posted this exact problem yesterday ("A curious identity, Part 2") , and Brian Charlesworth wrote an excellent solution... you may want to check it out.

If you want something a little different, you could ask for ∏ k = 1 2014 ( w k + 2 w − k ) \prod_{k=1}^{2014}(w^k+2w^{-k}) or ∏ k = 1 2014 ( w k − 2 w − k ) \prod_{k=1}^{2014}(w^k-2w^{-k}) ... the variations are endless... ;)

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Yes...there can be infinite variant. Wow ! these are truly curious identities :)

Pradeep Maurya - 6 years, 1 month ago

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