2070.41 – Stacking Grapefruit

Last week at the Farmer's Market, grapefruit were stacked in a triangular pyramid, 14 grapefruit on each bottom edge, 13 grapefruit on the next layer's edge, and so on up to the top where a single grapefruit sat in solitary splendor.

How many grapefruit were in the whole stack?

Solution

Starting at the top the layers look as in the figure below, part A. The numbers in each layer are

$1, 1 + 2, 1 + 2 + 3, \ldots, 1 + 2 + 3 . . . + 14.$

Added up these are $1, 3, 6, \ldots 105.$ For obvious reasons, these numbers are called the triangular numbers.

There is a formula for the triangular numbers that you can use to solve the problem:

$T_n = n\text{'th triangular number} = \frac{n (n - 1)}{2}.$

Or you can use Pascal's triangle as in the figure, part B. The third column gives the triangular numbers; the fourth column gives the $sum$ of the first $n$ triangular numbers. The solution of the problem is circled.

This is a very large number of grapefruit. Can you make a guess as to how tall the stack was? We hope that nobody dislodged a grapefruit in the bottom row.