Question 5.7.10: Pounds and Pounds of Silver As a reward for saving his kingd......
Pounds and Pounds of Silver
As a reward for saving his kingdom from a band of thieves, a king offered a knight one of two options. The knight’s first option was to be paid 100,000 pounds of silver all at once. The second option was to be paid over the course of a month. On the first day, he would receive one pound of silver. On the second day, he would receive two pounds of silver. On the third day, he would receive four pounds of silver, and so on, each day receiving double the amount given on the previous day. Assuming the month has 30 days, which option would provide the knight with more silver?
The first option pays the knight 100,000 pounds of silver. The second option pays according to the geometric sequence 1, 2, 4, 8, 16, . . . . In this
sequence, a_1 = 1, r = 2, and n = 30. The sum of this sequence can be found by substituting these values into the formula to obtain
s_n = \frac{a_1(1 – r^n)}{1 – r}
s_{30} = \frac{1(1 – 2^{30})}{1 – 2}
= \frac{1 – 1,073,741,824}{-1}
= \frac{ -1,073,741,823}{-1}
= 1,073,741,823
Thus, the knight would get 1,073,741,823 pounds of silver with the second option. Compared with the first option, the second would result in an additional 1,073,641,823 pounds of silver.