HMMT 十一月 2022 · GEN 赛 · 第 4 题

HMMT November 2022 — GEN Round — Problem 4

Let x < 0 . 1 be a positive real number. Let the foury series be 4 + 4 x + 4 x + 4 x + . . . , and let the 2 3 fourier series be 4 + 44 x + 444 x + 4444 x + . . . . Suppose that the sum of the fourier series is four times the sum of the foury series. Compute x .

解析

Let x < 0 . 1 be a positive real number. Let the foury series be 4 + 4 x + 4 x + 4 x + . . . , and let the 2 3 fourier series be 4 + 44 x + 444 x + 4444 x + . . . . Suppose that the sum of the fourier series is four times the sum of the foury series. Compute x . Proposed by: Carl Schildkraut, Luke Robitaille, Priya Ganesh 3 Answer: 40 4 Solution 1: The sum of the foury series can be expressed as by geometric series. The fourier 1 − x series can be expressed as 4 2 (10 − 1) + (100 − 1) x + (1000 − 1) x + . . . 9 4 2 2 = (10 + 100 x + 1000 x + . . . ) − (1 + x + x + . . . ) 9 4 10 1 = − . 9 1 − 10 x 1 − x Now we solve for x in the equation 4 10 1 4 − = 4 · 9 1 − 10 x 1 − x 1 − x 3 by multiplying both sides by (1 − 10 x )(1 − x ). We get x = . 40 Solution 2: Let R be the sum of the fourier series. Then the sum of the foury series is (1 − 10 x ) R . Thus, 1 − 10 x = 1 / 4 = ⇒ x = 3 / 40.