Question

Jerry plans to begin saving money for a trip by putting $1 in a savings account the first month and then adding enough to double the amount in the account each following month.

a. After he finishes contributing to the account, he will withdraw $500 to make a down payment on the trip. Create a function to show how much money will be in the account if he finishes contributing to the account after t months.

Answer 1

The amount he deposits is a geometric sequence:
1, 2, 4, 8, 16, ...

This is a geometric sequence with a first term of 1 and a common ratio of 2.
a = 1
r = 2

The general formula for the sum of the first t terms of a geometric series is:
S(t) = a(1 - r^t) / (1 - r)

Plug in your values for a and r:
S(t) = 1(1 - 2^t) / (1 - 2)
S(t) = (1 - 2^t) / -1
S(t) = -1 + 2^t
S(t) = 2^t - 1

Answer:
S(t) = 2^t - 1

P.S. If he did this for 9 months, he'd have saved $511.
P.P.S. If he did this for 2 years, he'd have $16,777,215.
P.P.P.S. If he did this for 3 years, he'd have the net worth of Bill Gates at around $70 billion.
P.P.P.P.S. If he did this for 4 years, he'd have about $281 trillion. That's enough to give each person in the U.S. about $1 million each.