The Twelve Days of Christmas
I guess that you know this Christmas song. If not, a simple search will solve it.
As you can read in music lyrics we have this case:
You receive in the first day:
A Partridge in a Pear Tree
In the second day you receive:
Two Turtle Doves and A Partridge in a Pear Tree
In the third day you receive:
Three French Hens, Two Turtle Doves and A Partridge in a Pear Tree
And so on until the twelfth day (look the image above the presents).
After these twelve days you will receive how many presents from your true love?
Image Source: Wikipedia
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There are 1 2 ∗ 1 + 1 1 ∗ 2 + . . . 1 ∗ 1 2 gifts in total.
So we can write that as sum of n*(13-n) for n between 1 and 12 inclusive. We need the sum
n = 1 ∑ 1 2 n ( 1 3 − n ) = 1 3 n = 1 ∑ 1 2 n − n = 1 ∑ 1 2 n 2 = 1 3 ∗ ( 1 2 ) ∗ ( 1 3 ) / 2 − ( 1 2 ) ∗ ( 1 3 ) ( 2 5 ) / 6 = 1 2 ∗ 1 3 ∗ ( 1 3 / 2 − 2 5 / 6 ) = 1 2 ∗ 1 3 ∗ ( 1 4 / 6 ) = 2 ∗ 1 3 ∗ 1 4 = 3 6 4