Suppose that the song "99 Bottles of Beer on the Wall" was sung from beginning to end.

What would be the sum of all the numbers (including repeats, of course) in the song?

Then, suppose you have some arbitrary number n bottles, which you would like to sing about. Find a formula in terms of n for the sum of all the numbers you will be singing.

Most verses of the song "99 Bottles of Beer on the Wall" follows the formulaic pattern from 99 down to 1:

NUMBER bottles of beer on the wall

NUMBER bottles of beer!

(You) take one down, and pass it around

(NUMBER - 1) bottles of beer on the wall!


The sum of each verse then is equal to 3 times the number of the first NUMBER mentioned in the verse. With the sum of 99 to 1 being 4950, and then multiplied by 3 [i.e, NUMBER + NUMBER + 1 + (NUMBER - 1) = 3 x NUMBER], the total rises to 14850.

Yet, 14850 is not the final total. Though there are many variations to the final verse (as there are also many variations to the song), the common ending is...

No bottles of beer on the wall!

No bottles of beer!

Go to the store and buy some more

99 bottles of beer on the wall!


...thus, there is an additional NUMBER of (99) bottles of beer to add, bringing the total to 14949.


If the song was for some arbitrary n number of bottles, with the final verse of n bottles, one such formula for the total number could be expressed as:
3/2(n² + n) + n.

 

Edited on September 5, 2008, 4:12 am

Posted by Dej Mar on 2008-09-05 02:53:48