Question

One of your household chores is to do the dishes every night after dinner.Your parents come to you with two choices about how you want to earn your allowance. Option one: is that you will be paid $250.00 at the end of every month for the work you do Option two: is that they will pay you a penny the first day and double your play each day for the rest of the month Which method of payment is the better deal? Use a problem solving strategy to determine your answer. Explain, in clear sentences, why your decision is the best one. Your explanation must prove why the method you chose is the better option

Answer 1

the penny option is the much better deal

Explanation:

The sequence of pay amounts for the "penny option" is ...

$0.01, $0.02, $0.04, $0.08 ...

For day n, the amount of pay is $0.01·2^(n-1). For day 30, the amount of pay is ...

$0.01·2^29 = $5,368,709.12

In case you can't tell, the "penny option" is a much better deal than $250.00. (You might need to credit check your parents before accepting this deal. I know my parents could not make good on this "penny option" offer.)

The total amount for the 30 days will be double this amount, less one cent, or ...

$10,737,418.23

_____

Sum of a geometric sequence

The daily payments of the penny option are a geometric sequence. The sum of n terms of such a sequence is given by ...

Sn = a1(r^n -1)/(r -1)

where a1 is the first term ($0.01) and r is the common ratio (2). In our case, the sum for n days is ...

Sn = $0.01·(2^n -1)

On day 15, this will already be more than $250. The accumulated pay on that day will be $0.01·(2^15-1) = $327.67.