Problem 2

Let

a_1,a_2,\ldots,a_n

be real constants and let

x

be real. Define

f(x)=\cos(a_1+x)+\frac12\cos(a_2+x)+\cdots+\frac1{2^{n-1}}\cos(a_n+x)

. Given that

f(x_1)=f(x_2)=0

, prove that

x_2-x_1=m\pi

for some integer

m

Step 2 of 4: Show the sinusoid is nonzero

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In plain words

The first term has weight larger than the total possible negative contribution of all later terms.

f(-a_1)\ge1-\sum_{j=2}^n2^{1-j}=2^{1-n}>0

Detailed analysis

At $x=-a_1$ , the first cosine equals $1$ . Every other cosine is at least $-1$ , while the remaining weights sum to $1-2^{1-n}$ . Hence $f(-a_1)\ge2^{1-n}>0$ , so $B$ and $C$ cannot both vanish.