When you were born, both of your rich grandparents made a deal with your parents, to do with what money they'll give you for birthdays. Grandad's deal was :

I'll give you £1 for your first birthday. Then, the amounts that I give you each year will double every time

and Grandma said :

I will start by giving you £1 for your first birthday. Then, the amount I give you will be the combined total of all the last birthday presents that I have ever given you up to that point. Plus, every birthday you have that's a multiple of five, I'll give you an extra five pounds.

a)What amount of money do they each pay you for your 15th birthday

and

Who would have given you the most after 27 years?

Let a(n) denote 2^(n-1) for n>=1 and let a(0)=1.

The amount of money that the grandfather gives on the n-th birthday is a(n). So on the 15th birthday, he gives 2^14=16,384. Also, the total payment from the grandfather over the first n years would be sum of a(k) over k=1 through n, which is (2^n)-1. So the payment over first 27 years would be (2^27)-1. I hope my grandfather was that rich and generous!

The payments from the gradmother are a bit more involved. It consists of a few streams; the initial payment starting on year 1, generates the same stream as the grandfathers, but lagged by one year, i.e., a(n-1) for n>=1. In addition each unit of the extra payment of 5 made on the birthyears ending in multiples of 5 (5,10,15,..) generate similar additional streams. To give a closed form for the grandmothers payment b(n) on year n, let n=5.m+k where 0<=k<5, then
b(n)=a(n-1) + 5.[sum of a(n-5j) where j varies from 1 to m.]

For instance, 15=5.3+0. So, b(15) = a(14)+5.[a(10)+a(5)+a(0)]=2^13+5.[2^9+2^4+1]
=10,837

Now the sum of a(0)+a(1)+…+a(k) = 2^k for any k>=1. Hence the cumulative payment made by grandmother for first n years, i.e. b(1)+b(2)+…+b(n), where n=5.m+k is
(2^(n-1))+5.[sum of (2^(n-5j) where j varies from 1 to m]
=(2^(n-1)) + 5.(2^k).[(2^5m)-1]/[(2^5)-1]

Thus, the total payment of the grandmother over the first 27=5*5+2 years is
(2^26) + 5.(2^2).[(2^25)-1]/[(2^5-1]. Again, I wish I had such a grandmother!!

Posted by Prab on 2003-11-06 15:19:09