Stumbling around on the complex plane

Probability Level 5

Let C C be the constant term of:

( x 1 + i + x 1 − i + x − 1 + i + x − 1 − i ) 14 (x^{1+\mathrm{i}}+x^{1-\mathrm{i}}+x^{-1+\mathrm{i}}+x^{-1-\mathrm{i}})^{14}

What is C \sqrt{C} ?


The answer is 3432.

Kerry Soderdahl by Kerry Soderdahl , 23, USA

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1 solution

Pulkit Gupta
Jan 10, 2016

We rewrite the equation as ( x i ( x + 1 x ) + 1 x i ( x + 1 x ) ) 14 = ( ( x + 1 x ) ( x i + 1 x i ) ) 14 \large (x^i ( x + \frac{1}{x} ) + \frac{1}{x^i} (x + \frac{1}{x}))^{14} = (( x + \frac{1}{x} ) ( x^i + \frac{1}{x^i}))^{14} .

Expanding both the terms binomially, we get identical constant terms 14C7 for both. Notice that the square of 14C7 will be the only constant term we get when we multiply the expansions together.

The question then simply asks for the value of 14C7 = 3432.

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