1. Series – Mortgage & Savings Question - You decide to buy a house and take out a mortgage for £60,000, at a fixed interest rate of 5% This 5% interest is calculated and added annually You pay off the same amount every year and you want to pay it all off after 25 years. How much do you need to repay per year?

2. Method: Let the amount of repayment per year be £P After 1 year, one year’s interest is calculated on £60,000 The amount owed is then........................ £60,000 x 1.05 After a repayment P amount owed is........ £60,000 x 1.05 - P After 2 years, one year’s interest is added and amount owed is...................... (£ 60,000 x 1.05 - P) x 1.05 Expand the brackets in your expression to get: £60,000 x 1.05 2 - 1.05P And the second years repayment P must be made giving:.......... £60,000 x 1.05 2 - 1.05P - P

3. After 3 years, after interest added, before repayment, amount owed is the end of second year total multiplied by 1.05 (60,000 x 1.05 2 - 1.05P – P) x 1.05 Expanding the brackets in this expression gives: 60,000 x 1.05 3 - 1.05 2 P – 1.05P After repayment P amount owed is : 60,000 x 1.05 3 - 1.05 2 P – 1.05P - P We should NOW be able to see a pattern emerging................ End of Year 1 - 60,000 x 1.05 - P End of Year 2 - 60,000 x 1.05 2 - 1.05P – P End of Year 3 - 60,000 x 1.05 3 - 1.05 2 P – 1.05P - P End of Year 25?? 60,000 ×1.05 25 – P ×1.05 24 – P ×1.05 23 -...................................- P×1.05 – P

4. This final repayment amount can be rewritten as 60,000 ×1.05 25 – (P ×1.05 24 + P ×1.05 23 +...................................+ P×1.05 + P) = 0 (P+ PX1.05 + P ×1.05 2 + P ×1.05 3 +..........................+ P×1.05 23 + Px1.05 24 ) This is a geometric series that we can find the sum of using So after 25 years the amount owed must be repaid and so: 60,000 ×1.05 25 = (P+ PX1.05 + P ×1.05 2 + P ×1.05 3 +................+ P×1.05 23 + Px1.05 24 ) This expression for total repayment must be equal to ZERO

5. And so... 60,000 ×1.05 25 = This we can solve for P We need to pay £4257.15 a year to clear the mortgage

6. Question 2 – You want to save to buy a car and decide you need about £7,000. You can afford £200 per month You find a savings account which gives 0.5% interest per month. How many months do you need to save for? Method- Again you need to find the general expression for the monthly amount saved After one month you have - 200 x 1.005 (remember to add interest) After two months you have - (200 x 1.005+200) x 1.005 Expanding the brackets 200 x 1.005 2 + 200 x 1.005 Factorising this expression 200(1.005 2 + 1.005)

7. So after n months you have : (recognising the pattern) 200 ×(1.005 n + 1.005 n-1 +.... + 1.005 2 + 1.005) = 200 ×1.005(1 + 1.005 +..1.005 n-1 ) This can be rewritten as : And so the above expression must be equal £ 7000 Note that 200 x 1.005 (1 + 1.005 +..1.005 n-1 ) is a geometric series that can be summed using So we can solve this for “n” But we need to use logs here!!

8. Simplify and rearrange... (step by step) Now we have it !! Take Logs So we need to solve this

9. So eventually n = 32.18 And so you need to save for 33 months to buy this