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Series and modelling for SQA Higher Maths

Series and modelling

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.

Modelling sequences

1

Andy borrows ยฃ1200 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)โฟ.
For n = 3, 1200(1.037)ยณ = ยฃ1338.19 (to 2 dp).

ยฃ1338.19

Andy borrows ยฃ1200 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.

2

A new cafรฉ makes a profit of ยฃ70 from selling coffees and ยฃ200 from selling food items. Given that the cafรฉ 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)ยฒ + 200(1.04)ยฒ = ยฃ290.58

ยฃ290.58

A new cafรฉ makes a profit of ยฃ70 from selling coffees and ยฃ200 from selling food items. Given that the cafรฉ 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.

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 ยฃ42000, calculate his salary after he completes 5 professional exams.

This is a geometric series of the form 42000(1.3)โฟ.
For n = 5, 42000(1.3)โต = ยฃ155943.06 (to 2 dp).

ยฃ155943.06

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 ยฃ42000, calculate his salary after he completes 5 professional exams.

4

Anna has a piggy bank, where she is putting money every week. Given that in the first week, she puts in ยฃ12 and the amount she puts in every week increases by ยฃ3, 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 ร— 12 + (6 โˆ’ 1)(3)) = ยฃ117.

ยฃ117

Anna has a piggy bank, where she is putting money every week. Given that in the first week, she puts in ยฃ12 and the amount she puts in every week increases by ยฃ3, calculate how much money there will be in the piggy bank after 6 weeks.

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 ยฃ25000, calculate their profit in their 4th year.

This is a geometric series of the form 25000(1.05)โฟ.
For n = 4, 25000(1.05)โด = ยฃ30387.66 (to 2 dp).

ยฃ30387.66

A business observes that their profit increases by 5% every year. Given that in the first year of operation, they made a profit of ยฃ25000, calculate their profit in their 4th year.

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