# Thread: Functions and Relations

1. ## Functions and Relations

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 you use the formula tn=ar^n-1 but im not exactly sure how.

2. Originally Posted by VDestinV
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 you use the formula tn=ar^n-1 but im not exactly sure how.
Lets write this out so...

$1(2)^0+1(2)^1+1(2)^2+...1(2)^{30}=\sum_{n=1}^{30}2 ^{n-1}$

The sum of the series will be

$S_{30}=a_1\frac{1-r^n}{1-r}=1\cdot \frac{1-2^{30}}{1-2}=2^{30}-1=1073741824 \mbox{cents}$

or in dollars

$$10,737,418.24$ a little over ten million The king made a bad choice 3. Actually I'm wrong that is for 31 days of labor. The sum should only go to 29 and should be 536870912 cents or$ 5,368,709.12 about 5 million dollars.

Good luck.