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Mathematics Extension 1 Course notes and theory Year 12, 2014 (Mr Vass)
Applications of Series (Syllabus reference 7.5 ) The principal focus of this unit is to link your prior knowledge of series with some common applications. One of the many applications of sequences and series occurs in financial mathematics. Here we will briefly discuss compound interest and superannuation.
For additional online assistance check out the Canvas course page where there are links to online lessons and resources such as: http://www.amsi.org.au/ESA_Senior_Years/SeniorTopic1/1_md/SeniorTopic1d.html#content_4
NOTE: All text book references in this unit of work are for Pender et al, Cambridge Year 12 3unit (Extension 1) and used under license by Newington College. Unless otherwise specified.
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Contents
Contents Syllabus Lesson 1 Review of sequences and series formulae
Theory APs Theory GPs Practice
Lesson 2 Compound interest Theory Further work Practice
Lesson 3 Annuities (e.g. superannuation) Theory Practice Further work
Lesson 4 Loan repayments Theory Practice Further work Assignment Challenge Exercise
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Syllabus
For further detail you can access the complete syllabus here on google drive: Syllabus (use your Newington google account to login)
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Lesson 1 - Review of sequences and series formulae
Theory - APs Arithmetic and geometric sequences were studied in Year 11 — this section will review the main results about APs and GPs and apply them to problems. Many of the applications will be financial, in preparation for the next three sections. Formulae for Arithmetic Sequences:
Examples Example 1
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Example 2
Theory - GPs Formulae for Geometric Sequences:
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Examples Example 3
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Example 4
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Practice Exercise 7A Pender
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Answers
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Lesson 2 - Compound interest
Theory Reviewing the formulae for simple and compound interest, we can see that both of these are just applications of sequences. Simple interest can be understood mathematically both as an arithmetic sequence and as a linear function. Compound interest or depreciation can be understood both as a geometric sequence and as an exponential function.
Substituting into this function the positive integers n = 1, 2, 3, . . . gives the sequence P+PR, P+2PR, P+3PR, . . . which is an AP with first term P+PR and common difference PR.
Substituting n = 1, 2, 3, . . . into this function gives the sequence P(1+R)1, P(1+R)2, P(1+R)3, . . . which is a GP with first term P(1 + R) and common ratio 1+R. NOTE 1: This formula only works when the compounding occurs over the SAME unit of time that the interest rate is given. NOTE 2: The proof below will need to be used in future to develop harder $$ problems.
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Examples Example 1
Example 2
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Example 3
Example 4 (Depreciation)
Further work These practice exercises are to be completed to further knowledge, skills and understanding in this concept area (any unfinished should be completed for HW):
Exercise 7B Questions 1,3,5,7,9,11,13 Text: Cambridge 3U Yr 12
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Practice (Source: M Grove, Maths in Focus, Extension 1)
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Answers
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Lesson 3 -Annuities (e.g. superannuation)
Theory An annuity is a fund where a certain amount of money is invested regularly (often annually, which is where the name comes from) for a number of years. Scholarship and trust funds, certain types of insurance and superannuation are examples of annuities. While superannuation and other investments are affected by changes in the economy, due to their longterm nature, the amount of interest they earn balances out over the years. In this course, we average out these changes, and use a constant amount of interest annually. These investment schemes, typically superannuation schemes, require money to be invested at regular intervals such as every month or every year. This makes things difficult, because each individual instalment earns compound interest for a different length of time. Hence calculating the value of these investments at some future time requires the theory of GPs. NOTE: This topic is intended to be an application of GPs, and learning formulae is not recommended. The most straightforward way to solve these problems is to find what each instalment grows to as it accrues compound interest. These final amounts form a GP, which can then be summed.
Examples Example 1
It is easier to keep track of each annual amount separately. The first amount earns interest for 25 years, the 2nd amount earns interest for 24 years, the 3rd amount for 23 years and so on.
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Example 2
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Example 3
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Practice (Source: M Grove, Maths in Focus, Extension 1)
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Answers
Further work These practice exercises are to be completed to further knowledge, skills and understanding in this concept area (any unfinished should be completed for HW):
Exercise 7C Questions 1 7, 11 Text: Cambridge 3U Yr 12
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Lesson 4 - Loan repayments
Theory Longterm loans such as housing loans are usually paid off by regular instalments, with compound interest charged on the balance owing at any time. The calculations associated with paying off a loan are therefore similar to the investment calculations of the previous section. As with superannuation, the most straightforward method is to calculate the final value of each payment as it accrues compound interest, and then add these final values up using the theory of GPs. We must also deal with the final value of the initial loan.
Examples Example 1
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Example 2
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Example 3
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Practice (Source: M Grove, Maths in Focus, Extension 1)
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Answers
Further work These practice exercises are to be completed to further knowledge, skills and understanding in this concept area (any unfinished should be completed for HW):
Exercise 7D Questions 1, 3, 5, 8, 12 Text: Cambridge 3U Yr 12
Assignment Challenge Ex 8 Hand in task
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Assignment - Challenge Exercise
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Answers
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