02.05 How is interest calculated?

Photo by Andre Taissin on Unsplash

Investing plans calculate interest on specific days. When planning how to invest your savings, you will want to compare plans based on not only the interest they provide, but whether they have a fixed interest rate for a specific time frame with a penalty for early withdrawal (like a bond) or whether the interest is variable with no penalty for withdrawals (like a savings account). To compare investing plans,  you will need to know approximately how much you will save, how often you will deposit into the account, and what the interest rate is. This calculation is easy if you understand arithmetic sequences.To learn more about how to recognize arithmetic sequences and how to calculate for a particular number in an arithmetic sequence, read one of the following websites:

• Arithmetic sequences on the Math is Fun website (summing an arithmetic sequence or series isn’t required, but is easy and interesting though it looks complicated. Do the ‘Your Turn’ at the bottom.)
• Arithmetic sequences on the Math Planet website (includes a video)
• Arithmetic sequences, including recursive sequences on OpenText BC website.

If you look at an arithmetic sequence equation, it looks like a linear function, which you would have studied in previous math classes. The main difference between the two is an arithmetic sequence is discrete (has set values) whereas a linear function is continuous. Remember that height, weight and age are all considered continuous values, but number of people are discrete values (because you can’t have a portion of a person).  If you want a more complete answer, you may choose to read this website comparing arithmetic sequences and linear functions.

An arithmetic sequence has the general rule:

t_n=a+(n-1)d where t_n is the n^th term, a is the first term, or t₁, and d is the common difference (the difference between each of the terms in the arithmetic sequence).

As a simple starting question, consider a $750 investment which, after a year in a savings account, was worth $783.75, after two years was worth $817.50, after three years $851.25 and after 4 years $885. What would be:

• the annual interest rate?
• the general term for the balance at the end of the nth year?
• the balance at the end of the 15th year?

Check your answers against the example of page 5-7 of this online textbook excerpt.

If you want, compare current interest rates on savings accounts, chequing accounts, and savings bonds and discuss with your parents which you should be using for your savings.

• Geometric sequences on the Math is Fun website
• Geometric sequences on the Math Planet website (includes a video)
• Geometric sequences, including recursive sequences on OpenText BC website.

For a clear comparison between geometric sequences and exponential growth (which we covered in 02.03), read through this online book and do the questions at the end. (approximately 10 minutes reading and 30 minutes of exercises)

Consider the difference between leasing and buying a car priced at $26,000. Compare a 3-year loan and a 3-year lease for this car. After 3 years, the car will be worth 50% of original value, or $13,000 — regardless of whether we lease or buy. Use a finance rate for both the loan and lease of 5.0% APR. Assume no down payment for either. How much will your monthly payment be for a loan? Remember when finding your formula, your monthly payment is decreasing your total amount, though you are also increasing the amount because of the interest.