Sums of a sequence: How many total gifts did my true love give to me during the entire 12 days of Christmas? Hint: Without any of that other stuff, there were 12 partridges in pear trees alone.

On the first day of Christmas,

my true love sent to me

A partridge in a pear tree. On the second day of Christmas,

my true love sent to me

Two turtle doves,

And a partridge in a pear tree. On the third day of Christmas,

my true love sent to me

Three French hens,

Two turtle doves,

And a partridge in a pear tree.

The song continues, adding 4 calling birds on the 4th day, 5 golden rings on the 5th, and so on up to the 12th day, when 12 drummers add to the cacophony of assorted birds, pipers and lords leaping all over the place.

Notice that on each day there is one partridge (so I will have 12 partridges by the 12th day), and each day from the second day onwards there are 2 doves (so I will have 22 doves), and from the 3rd there are 3 hens (total of 30 hens), and so on.

So, how many presents are there altogether?

Partridges: 1 * 12 = 12

Doves: 2 * 11 = 22

Hens 3 * 10 = 30

Calling birds: 4 * 9 = 36

Golden rings: 5 * 8 = 40

Geese: 6 * 7 = 42

Swans: 7 * 6 = 42

Maids: 8 * 5 = 40

Ladies: 9 * 4 = 36

Lords: 10 * 3 = 30

Pipers: 11 * 2 = 22

Drummers: 12 * 1 = 12

Total = 364

We observe that we have the same number of partridges as drummers (12 of each); doves and pipers (22 of each); hens and lords (30 of each) and so on. So the easiest way to count our presents is to add up to the middle of the list and then double the result: (12 + 22 + 30 + 36 + 40 + 42) * 2 = 364.