# Thread: Arithmetic and Geometric Sequences

1. ## Arithmetic and Geometric Sequences

Here is the question:

In an old story, a king asked a peasant to come and work for him. They both agreed that the job to be done could be completed in a month (30 days). The peasant asked the king if he could be paid in the following way: "On the first day that I work, I would like to be paid one penny. Each day that I work, I would like you to double my income." The king thought this was a great deal and happily agreed.

Analyze the deal that the king and the peasant made to determine whether or not the peasant made a good deal with the king.

Now I know that the peasant is getting the better deal. And I believe I have to use the formula tn=ar^n-1.

2. You're correct. The formula is $\displaystyle 2^{n}-1$

For instance, how much on the 5th day?. 16+8+4+2+1=31

$\displaystyle 2^{5}-1=31$

So, on the 30th day, he would have $\displaystyle 2^{30}-1$.

A rather large number, even in pennies.

That is \$10,737,418.23

The king may have him beheaded after seeing that.