## The Million \$ Mission -

### Patterns Galore

What if we wanted to calculate how much money that we'd make on a given day (say the 15th) without starting at the first day and calculating all the way up to it. If we could figure out a formula, then we could just plug 15 into it and crank it out. I bet you've already spotted lots of patterns from our calculating of the pay. Let's look at the first week again.

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

Look at the Pay for that Day Column. Do you notice a pattern there? Of course, you do. You're multiplying by 2 each time, so they are powers of 2. Let's write that column in cents rather than dollars, then when we finish we'll divide by 100 to get our answer in dollars. Let's also show the numbers as powers of 2. Look at the table below to see what I mean.

Pay with First Option - Week 1
Day No.
Pay for that Day (In Cents)
1
1 = 20
2
2 = 21
3
4 = 22
4
8 = 23
5
16 = 24
6
32 = 25
7
64 = 26

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

Pay for That Day Formula
Day No.
Pay for that Day (In Cents)
n
Any Day's Pay = 2n - 1
15
15th Day's Pay = 214

Let's put that in the calculator. Don't forget to divide by 100 to convert from cents to dollars, and we get \$163.84. Why don't you try one. Pick out a day; use this formula to calculate your pay for that particular day.

Find a formula for the Total Pay.
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