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Series and modelling for SQA Higher Maths
This page covers the following topics:
1. Modelling sequences
Sequences and series can be used to model real-life scenarios. When the model involves a change by a fixed amount, arithmetic sequences and series can be used. When the model involves a change by a factor or a percentage, geometric sequences and series can be used.
1
Andy borrows £1200 from the bank. Given that the annual interest on the loan is 3.7%, calculate the amount that Andy will owe to the bank in 3 years.
This is a geometric series of the form 1200(1.037)ⁿ.
For n = 3, 1200(1.037)³ = £1338.19 (to 2 dp).
£1338.19
2
A new café makes a profit of £70 from selling coffees and £200 from selling food items. Given that the café expects the profits from selling coffee to rise by 3% and the profits from food items to rise by 4% every week, calculate the profit made by the cafe in 2 weeks.
Profit = 70(1.03)² + 200(1.04)² = £290.58
£290.58
3
A bank employee receives a raise of 30% to their salary for every professional exam they complete. Given that Henry has a starting salary of £42000, calculate his salary after he completes 5 professional exams.
This is a geometric series of the form 42000(1.3)ⁿ.
For n = 5, 42000(1.3)⁵ = £155943.06 (to 2 dp).
£155943.06
4
Anna has a piggy bank, where she is putting money every week. Given that in the first week, she puts in £12 and the amount she puts in every week increases by £3, calculate how much money there will be in the piggy bank after 6 weeks.
The amount of money that there will be in the piggy bank is an arithmetic series.
Using the formula for an arithmetic series, money = (6/2)(2 × 12 + (6 − 1)(3)) = £117.
£117
5
A business observes that their profit increases by 5% every year. Given that in the first year of operation, they made a profit of £25000, calculate their profit in their 4th year.
This is a geometric series of the form 25000(1.05)ⁿ.
For n = 4, 25000(1.05)⁴ = £30387.66 (to 2 dp).
£30387.66
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