Hey, my name is Kanishk (student, grade9) I was wondering if you help me on this question: I recieve 1 penny the 1st day, 2 pennies the 2nd day, and my salary is doubled everyday for 30 days. How much money will I have by the end of the 30 day time period? (Is there a way of solving this problem without a chart?)



Hi Kanishk,

The total amount that you receive is

1 + 2 + 4 + 8 + 16 + ... where the sequence continues for 30 days. This is an example of a geometric series. A geometric series is constructed form a first term a and a number r, called the common ratio. The series is then
a + ar + ar 2 + ar 3 + ar 4 + ar 5 + ... In your series a = 1 and r = 2 and the number of terms is 30 so the sum is given by
S = a + ar + ar 2 + ar 3 + ar 4+ ... + ar 29

The key to finding a nice expression for this sum is to first multiply both sides by r to get

rS = ar + ar 2 + ar 3 + ar 4 + ar 5 + ... + ar 30 and then subtract rS from S.
S - rS
= a + ar + ar 2 + ar 3 + ar 4+ ... + ar 29 - ar - ar 2 - ar 3 - ar 4 - ar 5 - ... - ar 30
= a - ar 30 Hence
S = a (1 - r 30)/(1 - r)

Now evalute this expression when a = 1 and r = 2.

Cheers,
Penny
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