Problem:
The sum
2!1​+3!2​+4!3​+⋯+2022!2021​
can be expressed as a−b!1​, where a and b are positive integers. What is a+b?
Answer Choices:
A. 2020
B. 2021
C. 2022
D. 2023
E. 2024
Solution:
Notice that for n≥2,
n!n−1​=(n−1)!1​−n!1​.
Therefore the given sum telescopes:
1!1​−2!1​+2!1​−3!1​+3!1​−4!1​+⋯+2021!1​−2022!1​=1−2022!1​
Thus a=1,b=2022, and a+b=(D)2023​.
The problems on this page are the property of the MAA's American Mathematics Competitions