Question

The formula for the value of a finite geometric series ( with r # 1 ) is given as:

sn = a1 1-rⁿ/1-r
in where r is the common ratio, a1 is the first term in the series and Sn is the sum of the first n terms in the series. Answer the following problem by determining the values of a1,and Sn and implementing the above formula. Suppose that $ 1 was deposited into a bank account on the first day of September, $2 on the second day, $4 on the third day, and so on in a geometric sequence. Use your current Excel worksheet to determine the answers to the following:
How much money would be deposited at this rate on September 30 th?

Answer 1

By applying the formula for the sum of a finite geometric series, the total amount deposited in the bank account by the end of September 30th is $1,073,741,823, considering a starting deposit of $1 and a doubling pattern each day.

To solve this problem using the formula for the sum of a finite geometric series, we need to identify the values of
`\(a_1\), \(r\), and \(n\)`.

In this case:

-
`\(a_1\)` is the first term, which is $1.

- r is the common ratio, and from the problem, it's stated that each subsequent deposit is double the previous one, so r = 2.

- n is the number of terms, which is 30 (September has 30 days).

Now, we can use the formula:

`\[S_n = a_1 (1 - r^n)/(1 - r)\]`

Let's substitute the values:

`\[S_(30) = 1 * (1 - 2^(30))/(1 - 2)\]`

`\[=1 * (1 - 2^(30))/(1 - 2)\]\[S_(30) = 1 * (1 - 1073741824)/(1 - 2)\]\[S_(30) = 1 * (-1073741823)/(-1)\]\[S_(30) = 1073741823\]`

So, the total amount deposited by the end of September 30th in the described geometric sequence is $1,073,741,823.