Need help with arithmetic/geometric series

Some people believe they can make money from a chain letter (they are usually disappointed). A chain letter works roughly like this: A letter arrives with a list of four names attached and instructions to mail a copy to four more friends and to send 1 to the top name on the list. When you mail the four letters, you remove the top name (to whom the money was sent) and add your own name to the bottom of the list.

a) If no one breaks the chain, how much money do you receive?
b) Let d sub n be the number of dollars you receive if there are n names on the list instead of 4, but you still mailed to four friends. Find a formula for d sub n.

Please help me with this confusing problem. Also show your work and explain how you got to the answer. Thank you so much!

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Originally Posted by florx

Some people believe they can make money from a chain letter (they are usually disappointed). A chain letter works roughly like this: A letter arrives with a list of four names attached and instructions to mail a copy to four more friends and to send 1 to the top name on the list. When you mail the four letters, you remove the top name (to whom the money was sent) and add your own name to the bottom of the list.

a) If no one breaks the chain, how much money do you receive?

I assume you mean you send $1 or 1 pound or 1 Euro to the person at the top of the list. You send the letter to four people, and add your name to the bottom of the list. Each of those four send to each send to four more people, for a total of 16 people (assuming no duplicates) and adds their name, moving your name to third on the list. Each of those 16 sends out four letters to a total of 4(16)= 64 people and your name is moved to second on the list. Each of those 64 sends out 4 letters for a total of $4(64)= 4^5= 256$ and moves your name to first on the list. Each of those 256 sends you 1 (dollar or pound or Euro) so you would receive 256 (dollars or pounds or Euros).

Of course this is to good to be true and the reason such a scheme will not work (and is, in fact, illegal) is that
1) Since you send to people you know, they are likely to have friends in common and may send letters to the same people.
2) Many people will NOT forward such letters.
3) Most important- after a few iterations, the number of letters sent will be $4^{n+1}$ . For example, after 17 steps, that will be $4^18= 68719476736$ which is larger than the population of the world! According to the United Nations, in 2009 the entire population was 6,800,000,000. Everyone involved in the last few steps, will will be many times the number in the first steps will lose their money.

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b) Let d sub n be the number of dollars you receive if there are n names on the list instead of 4, but you still mailed to four friends. Find a formula for d sub n.

Since you, and each person to whom you send a letter, will send four letters, the number is multiplied by 4 each time. The number of times this repeats before you get your response is the number of times it take for your name to reach the top of the list which is n+1. 4 is multiplied by itself n+ 1 times giving $4^{n+1}$ people who are supposed to respond to you.

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Please help me with this confusing problem. Also show your work and explain how you got to the answer. Thank you so much!

Thank you so much for the explaining also. I forgot to add $1 but I guess you took it into consideration that it is currency that I am talking about. Sorry for that mistake.