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Need help with a proof... Please help! 10 pts!!?
Hey, I'm doing some review for my Combinatorics class at University... and I'm having trouble figuring out how to do this proof. Heres the question:
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Prove that, for every non-negative integer n,
[ ((1 - 3*i) / 2) * (1 + i)^n ] + [ ((1 + 3*i) / 2) * (1 - i)^n ]
is an integer.
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Please note that ' i ' means Complex (imaginary number). My prof said theres a few ways to solve this. I tried some induction but I got stuck. Any ideas? Thanks for any support!
1 Answer
- morningstarLv 71 decade agoFavorite Answer
I notice the real part of the left term is always equal to the real part of the right term, and the imaginary part of the left term is always equal to the negative of the imaginary part of the right term. Start out by trying to prove that.
Each successive increase of n multiplies the left term by i + 1 and the right term by i - 1. So for your inductive step you need to prove:
Given:
(a + bi)(1 + i) = c + di
Prove:
(a - bi)(1 - i) = c - di
Since the imaginary part of the left term equals the negative of the imaginary part of the right term, those parts cancel out. So at least you know the expression evaluates to a real number.
Since the real part of the left term equals the real part of the right term, then their sum is 2 times the real part of either term. Since the real part of each term is written as something over 2, then it seems plausible that twice that will come out to be an integer. You just need to prove the numerator is an integer. I guess I'll leave that up to you.
