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 she gives me 1 of something, 2 of something else, ..., of something else. At the end of the first 16 days, how many gifts has my true love given to me in total?
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Numerically, what is being added up here is 1 + 3 + 6 + 1 5 + . . . . . + 2 n ( n + 1 ) = 6 n ( n + 1 ) ( n + 2 ) . Plugging in 1 6 into this gives us the sum of the first 1 6 triangular numbers, which is 8 1 6 .