The Million \$ Mission -

Patterns Galore - 2

Could we also calculate our Total Pay without starting at the first day and calculating all the way up to it. We can if we can figure out a formula. We could then just plug 30 into the formula, and it could tell us how much we'd receive for all thirty days. Let's give it a try. Look at the chart for the first week again, and this time look at the Total Pay column.

Pay with First Option - Week 1
Day No.
Pay for that Day
Total Pay (In Dollars)
1
.01
.01
2
.02
.03
3
.04
.07
4
.08
.15
5
.16
.31
6
.32
.63
7
.64
1.27

Check the column again for a pattern. Hmmm. The numbers aren't powers of 2 this time - but they seem to be one less than powers of 2. (3 = 22 - 1 and 7 = 23 - 1.) So let's show the numbers as powers of 2 minus 1. Let's also write the column in cents rather than dollars, then when we finish we'll divide by 100 to get our answer in dollars. Look at the table below to see what I mean.

Pay with First Option - Week 1
Day No.
Total Pay (In Cents)
1
1 = 21 -1
2
3= 22 -1
3
7 = 23 -1
4
15= 24 -1
5
31= 25 -1
6
63= 26 -1
7
127= 27 -1

Do you see a relationship between the Day No. and the exponent? If you do, then again we have a formula where we could just substitute in the Day No. and get the Total Pay up to that day.

Total Pay Formula
Day No.
Total Pay (In Cents)
n
Any Day's Total Pay = 2n- 1
30
30th Day's Pay = 230-1.

Let's put that in the calculator. Don't forget to divide by 100 to convert from cents to dollars, and we get \$10,737, 418.23. Why don't you try one? Pick out a day; use this formula to calculate your total for that particular day.

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