By: Tao Steven Zheng (郑涛)

【Problem】

The following problem is attributed to Alcuin of York (c. 735 – 804 AD):

 Suppose there is a stairway which has 100 steps. One pigeon sat on the first step, two pigeons on the second step, three pigeons on the third step, four on the fourth step, five on the fifth step, and so on up to the hundredth step. How many pigeons are there altogether?

【Solution】

Let   denote the sum of consecutive positive integers. Determine the formula for calculating this sum using the arithmetic series formula. The first term is , and the final term is . Hence,

The total number of pigeons is the sum of the first 100 positive integers.

Therefore, there are 5050 pigeons on the stairway.