One man put 1 coin in a pot on the first day. The next visit he put 2 gold coins. He continued this for 30 visits, always doubling the number of coins.

Question

How many coins did he put in the pot on the 30th day? nth day?

How many gold coins are in the pot in total after 30 visits? nth visits?

Answer 1

This is basically a question about a geometric series with first term 1 and ratio 2:

 

1 + 2 + 2² + 2³ + ... + 2ⁿ .

 

There are two important formulas which will be helpful here:

 

S_n = a(1 - rⁿ)/(1 - r) is the sum of the first n terms in the series. a is the first term and r is the ratio.

 

a_n = a·r^n-1 is the n^th term.

 

How many gold coins are put in the pot on the 30th day?

 

The answer is the number a₃₀ = 1·2^30-1 = 2²⁹ = 536,870,912 gold coins.

 

How many are put in the pot on the n^th day?

 

The answer is a_n = 1·2^n-1 = 2^n-1.

 

How many gold coins are in the pot after 30 days?

 

The answer is S₃₀ = 1(1 - 2³⁰)/(1 - 2) = 2³⁰ - 1 = 1,073,741,823 gold coins.

 

How many are in the pot after n days?

 

The answer is S_n = 1(1 - 2ⁿ)/(1 - 2) = 2ⁿ - 1.