Let be real constants

MATHEMATICS · Subjective · A Level · Mathematics · Proof · Proof

Problem

Let

a_1, a_2, \dots, a_n

be real constants, and

x

be a real variable, and

f(x) = \cos(a_1 + x) + \frac{1}{2}\cos(a_2 + x) + \frac{1}{4}\cos(a_3 + x) + \dots + \frac{1}{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