Emile got a letter in the mail saying that a wealthy relative had left him an inheritance. At a meeting the next week, a lawyer read the following statement.
To Emile, because he likes math problems, I leave a choice. He can have one of two inheritances. He must make his choice before leaving the office today.
Option 1: A starting amount of $10,000 and then, for the next 26 years, an annual payment at the end of each year that is $1000 more than the amount he received the previous year.
Option 2: A starting amount of one cent and then, for the next 26 years, an annual payment at the end of each year that is twice the amount he received the previous year.
(For each option, there is a total of 27 payments) Luckily, Emile brought his calculator to the meeting.
Suppose f(n) is the amount of money Emile gets in Year n under Option 1 and
g(n) is the amount of money he gets in Year n under Option 2.
Year 0 is the year in which Emile gets his first payment.
a. Find closed-form rules for f(n) and g(n).
b. For what values of n is f(n) < g(n).
c. For what values of n is f(n) > g(n).
Option 1 is the following sequence:
And so on.
Since this is an arithmetic sequence, we use the formula
f(n) = a + (n - 1) d, where a = 10 000 (amount in Year 1) and d = 1 000 (amount added each year)
f(n) = 10 000 + (n - 1) ...
Finding the best choice in inheritances. Answered in 233 words. A graph and formulas are provided.