And a partridge in a pear tree

Level 1

As the well-known song goes, on the first day of Christmas my true love gave to me a partridge in a pear tree. On the second day of Christmas, my true love gave to me 2 turtle doves and a partridge in a pear tree. On day n n she gives me 1 of something, 2 of something else, ..., n n of something else. At the end of the first 16 days, how many gifts has my true love given to me in total?


The answer is 816.

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2 solutions

Keshav Ramesh
Nov 22, 2017

Numerically, what is being added up here is 1 + 3 + 6 + 15 + . . . . . + n ( n + 1 ) 2 = n ( n + 1 ) ( n + 2 ) 6 1+3+6+15+.....+\frac{n(n+1)}{2}=\frac{n(n+1)(n+2)}{6} . Plugging in 16 16 into this gives us the sum of the first 16 16 triangular numbers, which is 816 816 .

Josh Rowley
Dec 24, 2013

In effect we have to look at 16 summations: 1, 1+2, ... , 1+2+....+16

The formula for summing the first n integers is n(n+1)/2. We should now also note that the formula for n choose 2 is n(n-1)/2. Clearly these 2 are comparable if we offset the ns by 1. So we have: 2C2 + 3C2 + 4C2 + ...... + 17C2

By the hockey stick identity this is in fact 18C3 = 816

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