Lesson_slides_-_3F_Modelling_sequences_using_a_rule__Edrolo.pptx

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1. Don’t believe everything Einstein ‘said’ ‘Hello’ © Michael Palmer & Edrolo 2023 Albert Einstein once said that ‘compound interest is the most powerful force in the universe’ 1

2. VCE GENERAL MATHEMATICS UNITS 1&2 CHAPTER 3: NUMBER PATTERNS AND RECURSION Presented by Michael Palmer Modelling sequences using a rule

3. Simple interest examples 2 Compound interest examples 3 Finding a single term 1 [ generation of the explicit rule, , of an arithmetic or geometric sequence 1 ] , its use and evaluation, including [ various practical and financial contexts 2 , 3 ]  

4. Finding a single term So far, we have been generating sequences of things like: and We’ve been thinking about these terms as having symbols: While there is no problem generating the terms of a sequence one by one, sometimes we wish to know the value of a single term. That is, I might like to know how to get the 38 th , or 102 nd term straight away. Let’s derive the formulas to do this.   Introduction

5. Finding a single term The recurrence equation for an arithmetic sequence is: If we start generating the sequence, then we get:   Finding a single term from an arithmetic sequence Find a single term from an arithmetic sequence (formula)  

6. Finding a single term Find a single term from a geometric sequence (formula)   The recurrence relation for a geometric sequence is: Generating the terms in the sequence, we get:   Finding a single term from a geometric sequence Teacher’s tip! Remember! When we multiply numbers with the same base, we add the powers.  

7. Find the term of the sequence:   Find a single term from an arithmetic sequence (formula)  

8. Find a single term from a geometric sequence (formula)   Find the term of the sequence:  

9. Find in this sequence:   I don’t know.   Find a single term from an arithmetic sequence (formula)  

10. Find in this sequence:   I don’t know.   Find a single term from an arithmetic sequence (formula)  

11. Find a single term from a geometric sequence (formula)   If , find .   I don’t know.  

12. If , find .   I don’t know.   Find a single term from a geometric sequence (formula)  

13. Simple interest examples In an earlier video, we used a recurrence relation to model the growth of a simple interest investment. Last time, we generated all the terms of the sequence. However, we can solve problems without having to find multiple terms. Introduction Key takeaway Find a single term from an arithmetic sequence (formula)   If we model a simple interest investment with the following recurrence relation: …then we can find a single term with the equation:  

14. A simple interest investment of $10 000 pays 7% p.a. State a suitable recurrence relation. Find the value of the investment after 10 years.

15. A simple interest investment of $20 000 pays 3% p.a. State the recurrence relation that models this situation and find when the value of the investment exceeds $24 000.

16. A simple interest investment of $30 000 pays 4% p.a. How many years will it take for the investment to exceed $40 000? 8 years 8.3333…years 9 years 10 years I don’t know.

17. A simple interest investment of $30 000 pays 4% p.a. How many years will it take for the investment to exceed $40 000? 8 years 8.3333…years 9 years 10 years I don’t know.

18. Compound interest examples The recurrence relation for compound interest is: A single term from the recurrence relation can be found using the formula:   Key takeaway Deep dive Find a single term from a geometric sequence (formula)   It’s important to note that each term is not necessarily separated by one year. The terms will be separated by different time intervals, which depend on how frequently the interest is calculated. If the interest compounds monthly, then there will be 12 terms in one year. If the interest compounds daily, then there will be 365 terms in one year.

19. An investment of $5000 pays interest at 4% p.a. compounding annually. The recurrence relation is: Find the value of the investment after 5 years. Find how long it will take for the value of the investment to exceed $10 000.   Find a single term from a geometric sequence (formula)  

20. An investment of $5000 pays interest at 4% p.a compounding monthly. The recurrence relation is: Find the value of the investment after 5 years. Find how long it will take for the value of the investment to exceed $10 000.   Find a single term from a geometric sequence (formula)  

21. An investment of $5000 pays interest at 4% p.a compounding daily. The recurrence relation is: Find the value of the investment after 5 years. Find how long it will take for the value of the investment to exceed $10 000.   Find a single term from a geometric sequence (formula)  

22. An investment of $20 000 pays interest at 4% p.a. compounding monthly . Find the value of the investment after 2 years, rounded to the nearest dollar. $20 134 $20 800 $21 663 $80 000 I don’t know. Find a single term from a geometric sequence (formula)  

23. An investment of $20 000 pays interest at 4% p.a. compounding monthly . Find the value of the investment after 2 years, rounded to the nearest dollar. $20 134 $20 800 $21 663 $80 000 I don’t know. Find a single term from a geometric sequence (formula)  

24. An investment of $20 000 pays interest at 4% p.a. compounding monthly . Find how long it will take for the value of the investment to exceed $24 000. 55 months 54 months 5 months 4 months I don’t know. Find a single term from a geometric sequence (formula)  

25. Find a single term from a geometric sequence (formula)   An investment of $20 000 pays interest at 4% p.a. compounding monthly . Find how long it will take for the value of the investment to exceed $24 000. 55 months 54 months 5 months 4 months I don’t know.

26. Summary The formula for finding a single term from an arithmetic sequence is: The formula for finding a single term from a geometric sequence is: When making calculations about compound interest, remember to adjust the formula when there are more compounding periods.   Key takeaway What’s coming next Chapter 3 – Progress check 1

27. Image credits and question sources Image attribution: Pages 1, 26 : Image by ParentRap / Pixabay.com license Question sources: All questions are written by Michael Palmer.

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