Lesson: Geometric Series

In this lesson, we will learn how to derive the formula to calculate any geometric series and how to use it to solve real-world problems.

Sample Question Videos

  • 03:24
  • 01:48

Worksheet: 25 Questions • 2 Videos

Q1:

To calculate the amount of money in a structured savings account, where a saver deposits a regular amount at regular time intervals, we consider each month’s deposit separately.

Consider a saver who makes a regular deposit on the last day of every month in an account where the interest is calculated on the last day of every month.

Let the regular deposit be 𝐷 , and let the monthly interest rate be 𝑖 (an interest rate of 𝑝 % would give an 𝑖 value of 𝑝 1 0 0 ).

On the day the 𝑛 th deposit is made, the first deposit has been earning interest for ( 𝑛 βˆ’ 1 ) months, so its value is 𝐷 ( 1 + 𝑖 )    .

Similarly, on the day the 𝑛 th deposit is made, the second deposit has been earning interest for ( 𝑛 βˆ’ 2 ) months, so its value is 𝐷 ( 1 + 𝑖 )    .

The pattern continues until we consider the 𝑛 th deposit which has earned no interest on the day it is deposited so its value is 𝐷 .

To calculate the total amount in the fund, 𝑇 , on the day the 𝑛 th deposit is made, we need to find the sum of the values of the individual deposits.

Starting with the 𝑛 th deposit, we get 𝑇 = 𝐷 + 𝐷 ( 1 + 𝑖 ) + 𝐷 ( 1 + 𝑖 ) + β‹― + 𝐷 ( 1 + 𝑖 ) .    

What kind of series do you see on the right-hand side of the equation?

Using the formula for the sum of the first 𝑛 terms of a geometric series, write a formula for 𝑇 , the total amount in the fund.

Q2:

Sarah saves $ 5 0 every month in a high-performing investment fund. The fund is guaranteed to pay 6 % annual interest, compounded monthly.

How much is Sarah guaranteed to have in her fund at the end of 2 years?

Q3:

Nader saves $ 2 0 every month in an account that pays an annual interest rate of 4 % compounded monthly.

How much will be in Nader’s account after 4 years of regular saving? Give your answer to the nearest cent.

If the interest was compounded quarterly, how much would be in the account after 4 years?

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