Question

If your employer asked you to select one of these options: Option 1: Your salary for the next two months will be $5,000. Option 2: 1 penny for the first day; then, he will double your salary every day for the next two months. Which would you pick and how much will your salary be at the end of the first month?

Answer 1

Option 2 is the best: 2^0 = 1

Pay after 1 month (Assuming the month has 30 days):

2^29/100 = $5,368,709.12

$5,368,709.12 definitely beats $5000 for the first month.

Answer 2

You pick the second option.

Your salary at the end of the first month will be $10,737,418.23

Explanation:

Option 1:

$5000 for the next two months.

Option 2:

A geometric sequence, with common ratio r = 2.

The common ratio of a geometric sequence is the division of the term
`a_(n+1)` by the term
`a_(n)`.

Here, the geometric sequence is {0.01, 0.02, 0.04,....}, since a penny is 1 cent.

The sum of the first n terms of a geometric sequence is given by the following formula:

`S_(n) = (a_(1)*(1 - r^(n)))/(1 - r)`

In which
`a_(1)` is the first term, so
`a_(1) = 0.01`.

For the next two months, so 60 days.

`S_(60) = (0.01*(1 - 2^(60)))/(1 - 2) = 1.15 * 10^(16)`

This is higher than $5,000, so you pick the second option.

How much will your salary be at the end of the first month?

`S_(30) = (0.01*(1 - 2^(30)))/(1 - 2) = 10,737,418.23`

Your salary at the end of the first month will be $10,737,418.23