jmc

Question

A certain collection of numbered indexed cards includes one card with a 1 written on it, two cards with a 2, and so forth up to n cards showing an n, for some positive integer n. Determine n, if the average value of a card in this collection is 2017.

Accepted Answer

The number of cards is 1 + 2 + 3 + + n = n(n + 1)2, and the sum of the values of all cards is 1^2 + 2^2 + 3^2 + + n^2 = n(n + 1)(2n + 1)6.Therefore, the average value of a card is n(n + 1)(2n + 1)6n(n + 1)2 = 2n + 13.Setting this to 2017 and solving, we find n = 3025.