## 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.