Google is offering a $500 initial payment and then will double their payment each month.

Question

I'm unaware on how to turn it into an equation

Answer 1

The "initial payment" occurs in month=0   [so, let m=month]

 

Now, in a positional numbering system (like decimal or binary), the value of each place going left is multiplied by the number base.   

 

That is, a decimal number is:

       a*10⁵ + b*10⁴ + c*10³ + d*10² + e*10¹ + f*10⁰           [note: 10⁰ = 1, so exponent is m]

 

A binary number is:

       a*2⁵ + b*2⁴ + c*2³ + d*2² + e*2¹ + f*2⁰

 

If the payment is doubled each time, the payments will each be:     500*2^m

 

After 5 months, the total is:

        500*2⁵ + 500*2⁴ + 500*2³ + 500*2² + 500*2¹ + 500*2⁰

    = 500*(2⁵ +2⁴ + 2³ + 2¹ + 2⁰⁾

    = 500*(2⁶ - 1)          [note that this is 2^m+1 - 1]

 

Answer 2

Let m = the number of months.

 

The initial payment is $500 (month 0). Month one is $500(2). Month 2 is $500(2)(2), etc.

 

f(m) = 500(2)^m

 

Again, at month 0, that's $500(2)⁰ = $500(1) = $500. Month 5 would be $500(2)⁵, etc.