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 above, part $A$ . The numbers in each layer are

$1, 1 + 2, 1 + 2 + 3, \cdots, 1 + 2 + 3 + \cdots + 14$

Added up, this sequence is $1, 3, 6, \ldots, 105.$ 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:

$T_n = n^\text{th} \text{ triangular number} = \dfrac{n (n - 1)}{2}$

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.