Brian deposited 1 cent into an empty non interest bearing bank account on the first day of the month. He then additionally deposited 3 cents on the second day, 9
cents on the third day, and 27 cents on the fourth day. What would be the total amount of money in the account at the end of the 20th day if the pattern
continued

Brian deposited 1 cent on the first day with an additional 3 cents, then 9 cents on the second day, 27 cents on the third day and so on, this form a geometric series.

What is a geometric series?

A geometric series is a series which has a constant ratio between consecutive terms. This ratio is called the common ratio, r.

Now, the total amount of money, since we add up till the 20 th day, is the geometric series,

s = 1 + 3 + 9 + 27 + ...

s = 1 + 3¹ + 3² + 3³ + ... + 3²⁰

Now the common ratio of the series is 3.

Sum of geometic series

Now the total sum is the sum of the series which is

S = a(rⁿ - 1)/(r - 1) where

a = first term = 1,
r = common ratio = 3 and
n = number of terms = 21

So, substituting the values of the variables into the equation, we have

S = a(rⁿ - 1)/(r - 1)

S = 1(3²¹ - 1)/(3 - 1)

S = (3²¹ - 1)/2

S = (10460353203 - 1)/2

S = 5230176601 cents

So, the total amount of money in the account at the end of the 20th day is 5230176601 cents

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