How do you calculate the number of cans in this arithmetic progression?

Question

Cans are arranged in a pile such that each row has one less can than the row below. There is a total of 3240 cans in the pile. How many cans are there in the bottom row?

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Answer 1 · 2018-01-05T23:24:04

#80#

If there is one can in the top row and there are #n# rows, then there are #n# cans in the bottom row and the total number of cans would be:

#1/2 n (n+1)#

So let's try to solve:

#3240 = 1/2 n(n+1)#

Multiply both sides by #2# to get:

#6480 = n(n+1)#

Note that #6480 = 6400+80 = 80^2+80 = 80(80+1)#

So there are #80# rows of cans with #80# in the bottom row.