Together, we will look at financial applications of sequences and series and then tackle real-world financial decisions like home loans, savings accounts, car loans, and superannuation. Consider this your Band 6 study guide to ace the financial maths segment of your exam.
Financial Applications of Sequences and Series
Before we delve into financial applications, let's recap the basics.
A sequence is a list of numbers arranged in a specific order, while a series is the sum of the terms in a sequence.
Using Geometric Sequences for Financial Analysis
Geometric sequences are handy for modelling situations involving exponential growth and decay, which can be seen in real-world applications such as finance. For instance, consider investments, loans, and savings accounts.
Key Idea:
In financial applications, the common ratio (r) represents the growth or decay factor, usually expressed as a decimal.
Example Question 1 - Savings Account
Sam invests $5,000 in a savings account with an annual interest rate of 4%. How much will he have in the account after 5 years?
N.B The initial $5000 is referred to as the principal (P)
Example Question 2a - Car Loan
Suppose Rachel takes out a car loan with a 12% annual interest rate, and you plan to make monthly payments of $300. How many months will it take to pay off a $15,000 car loan?
Example Question 2b - Car Loan
Myles takes out a loan of $34000 to buy a car. He will repay the loan in five years, paying 60 equal monthly payments, beginning a month after he takes out the loan. Interest is charged at 6% per annum, compounded monthly. Find how much the monthly instalment should be, to the nearest cent.
Example Question 3 - Home Loan
Liam takes out a loan of 200000 on the 1st of January 2016 to buy a house. He will repay the loan in monthly instalments of $2200. Interest is charged at 12% p.a., compounded monthly.
- Find a formula for the amount owing at the end of n months.
- How much will he owe after 5 years?
- How long does it take to repay
i) Half the loan?
ii) The full loan?
N.B: 12% p.a. = (0.12/12) = 0.01 = 1% per month = therefore the rate of interest used should be 1.01.
Example Question 4 - Superannuation
Tom invests $250 at the end of each month into his superannuation fund at 6% per annum, starting in 2007 and making his last payment in December 2045. How much will he have at the end of 2045.
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