Find the equation for compound (?) interes

Question

Here's the problem:

 

Stacey owes Carmen $20.

The First week, Stacey will have to pay an extra dollar.

The second week, Stacey will have to pay an extra $2.00

The third week, Stacey will have to pay an extra $4 (Her late fee doubles every week added on to the original $20).

 

What will she owe if she pays Carmen on the nth week?

Answer 1

Dan:

 

Assume that Stacey begins payments after one week and part of a week counts as a whole week.

 

First week           $  1

Second week       $  2

Third week          $  8

Fourth week        $16

 

Question:  What will Stacey owe as late fee on the n-th week? (Her late fee doubles every week added on to the original $20).
 

 

Confusion: The word "extra" is ambiguous.  Extra TO WHAT? 

 

To (a) the originally borrowed $20? OR 

 

Extra to (b) the amount already owed? 

 

For example, in the third week, does Stacy pay 20 + 4, OR, for the third week, she pays 20+ 1 + 2 + 4?

 

If (a), Stacey owes (in dollars) (20 + 2^{n - 1}).

 

If (b), Stacey owes(in dollars)  (20 + 1 + 2 + ... + 2^n-1) = 20 + 2ⁿ - 1

 

Makes a whopping difference as weeks go  by.  For example, if n= 6, from (a) we get 20+32=52. From (b) we get 20+64-1 = 83.

 

Dattaprabhakar (Dr. G.)

 

P.S. The proof of 1+2+.....+2^n-1 = 2ⁿ -1 (n > 1) may not be in your curriculum.  If you absolutely want to know it, post a comment for me.  OR see the web-link

 

http://math.stackexchange.com/questions/22599/how-do-i-prove-this-by-induction-sum-of-powers-of-2