Track / Stage 5 / 2 of 400 #602 of 1964

Problem 602

AIME late

Algebra Difficulty 5.0 Find the answer

Find all solutions to the equation $2017^{x}-2016^{x}=1$ .

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

5. It is obvious that $x=1$ will be a solution to the given equation. Let's show that there are no others. We have: $\left(\frac{2017}{2016}\right)^{x}=1+\frac{1}{2016^{x}} ;\left(\frac{2017}{2016}\right)^{x}-1=\frac{1}{2016^{x}}$ From this, it is clear that the function on the left side is increasing, while the function on the right side is decreasing. Therefore, the equation has no more than one solution. Answer: $\{1\}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.