Module 2 for Mathematics Advanced (HSC)

NOTE: This guide assumes that you fully understand the principles of the preliminary Advanced mathematics course.

## Simple and Compound Interest

• Simple interest $(I)$ is calculated with $\color{orange}I=PRn$, an Arithmetic Progression
• $P$ is the principal (initial) sum,
• $R$ is the rate of interest per unit of time
• $n$ is the number of time intervals which have passed
• If the question asks for the total amount, add $P$ to $I$ at the end
• Compound interest is found by $\color{orange}A_{n}=P(1+R)^{n}$, a Geometric Progression
• $A_n$ is the amount of interest after $n$ units of time
• To find the interest (without the initial amount), subtract $P$ from $A_n$
• Depreciation is a form of compound in terest, where the value decreases over time
• Depreciation is expressed as $\color{orange}A_{n}=P(1-R)^{n}$ (also a Geometric Progression)
• $R$ is the rate of depreciation per unit time
• To find the interest (without the initial amount), subtract $P$ from $A_n$

## Annuities

• Annuities are compound interest investments, from which equal payments are recieved on a regular basis, for a fixed period of time

#### Practice Question

Minho deposits 200$per month at the start of each month into an annuity which pays 6% p.a. for 20 years. How much will the account hold after the full 20 years? Toggle Answer • After 1 month, the account has$200(1+0.005)$dollars • After 2 months,$200(1.005)^{2}+200(1.005)$• After$n$months, we have$200(1.005^{n}+1.005^{n-1}+…+1.005)$• The geometric progression in the brackets is: $$S_{(20\times 12)}=\frac{1.005(1.005^{240})-1}{1.005-1}=464.3511$$ • Therefore,$464.3511\times 200=92870.22 after 20 years

## Present and Future Values

• The Future value $(FV)$ is the total value of an investment at the end of its term, including all interest
• The Present value $(PV)$ is the single lump of money that could be initially invested to yield a given future value over a given period
• Present values are calculated using the compound interest formula
• Future value is calculated using a variant of the compound interest formula: $$\color{orange}FV=PV(1+r)^{n}$$

## Loan Repayments

• Loans are usually repaid through regular installments, with compound interest charged on the balance owed
• $\color{orange}A_n = \text{principle + interest - installments + interest}$
• The loan is paid off when $A_{n}=0$

#### Practice Question

Michael takes out $10000 to buy a car. He will repay the loan in 5 years, paying 60 equal monthly instalments, beginning 1 month after he takes out the loan. Interest is 6% p.a. compounded monthly. How much is the monthly installment? Toggle Answer Method 1: Let$M$be the monthly installment: •$A_{1}=10000(1.005)-M$•$A_{2}=(10000(1.005)-M)(1.005)-M$•$\therefore A_{2}=10000(1.005)^{2}-1.005M-M$•$A_{60}=0=10000(1.005)^{60}-M(1+1.005+…+1.005^{59})$GP inside the brackets is$\frac{10000(1.005^{60})}{\frac{1.005^{60}-1}{0.005}}=\$193.33$

Method 2 (Speed Hack):

• $A_{n}=10000(1.005)^{n}-M(1+1.005+…+1.005^{n-1})$
• $10000(1.005)^{60}=M(1+1.005+…+1.005^{59})$

GP inside brackets is $S_{60}=\frac{1.005^{60}-1}{0.005}=69.77$

• $\therefore M=\frac{10000(1.005)^{60}}{69.77}$
• $=\$193.33 \$

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