General Maths Unit 1: Sequences and Finance Study Notes

Number Sequences and Patterns (Chapter 3A)

• Definition of a Sequence: A sequence is any list of terms (which can be numbers or symbols) presented in a particular, non-random order.

• Naming Conventions: Because terms have fixed places, they are named using the notation $t_n$, where $n$ represents the term number or the number of iterations of a rule.

  • Term 0 ($t_0$): This is known as the 'starting term' or 'initial term'.

  • Example: In the list $4, 8, 98, 67, 23\dots$:

    • $t_0 = 4$

    • $t_1 = 8$

    • $t_2 = 98$

    • $t_3 = 67$

Types of Sequence Generation

• Random Generation: Sequences can be produced without a rule, such as throwing a fair die: $1, 6, 3, 2, 3, 5\dots$

• Recursive Generation: Sequences generated using a mathematical rule. A common example is the results of multiplication tables, where a fixed difference is repeatedly added.

  • Example for the 3 times table: $3, 6, 9, 12, 15\dots$

  • Rule: Start at 3 ($t_0 = 3$), and the next term is the current term plus 3 ($t_{n+1} = t_n + 3$).

Observable Behaviors of Number Sequences

In General Mathematics, sequences are categorized by five main behaviors. A single sequence may exhibit more than one behavior.

1. I - Increasing: Each successive term has a greater value than the previous one.

   • Example: $1, 2, 3, 4, 5\dots$ ($t_0 = 1, t_{n+1} = t_n + 1$).

2. II - Decreasing: Each successive term has a smaller value than the previous one.

   • Example: $5, 4, 3, 2, 1\dots$ ($t_0 = 5, t_{n+1} = t_n - 1$).

3. III - Constant: Every successive term has the same value.

   • Example: $1, 1, 1, 1, 1\dots$ ($t_0 = 1, t_{n+1} = t_n$).

4. IV - Oscillating: The values of successive terms alternate or change between two or more values.

   • Example: $1, -1, 1, -1, 1\dots$ ($t_0 = 1, t_{n+1} = -t_n$).

5. V - Limiting: The sequence approaches a specific fixed value.

   • Example: $1, 0.1, 0.01, 0.001, 0.0001\dots$ ($t_0 = 1, t_{n+1} = 0.1 \times t_n$).

Examples of Identifying Sequence Behavior

• Example 1: $8, 4, 8, 4 \dots$

  • Defined as Oscillating.

  • Predicting $t_4$: Since $t_0=8, t_1=4, t_2=8, t_3=4$, the next term is $t_4 = 8$.

• Example 2: $65, 70, 75, 80$

  • Behavior: Increasing.

  • Rule: Add 5 to each term.

  • Recurrence Relation: $t_0 = 65, t_{n+1} = t_n + 5$.

  • Finding $t_5$:

    • $t_4 = 80 + 5 = 85$

    • $t_5 = 85 + 5 = 90$

• Example 3: $1000, 200, 40 \dots$

  • Behavior: Decreasing and Limiting (approaching zero).

  • Rule: Divide by 5 (which is the same as multiplying by $0.2$).

  • Recurrence Relation (written as multiplication): $t_0 = 1000, t_{n+1} = 0.2 \times t_n$.

  • Finding $t_3$: $t_3 = 40 \times 0.2 = 8$.

• Example 5 (Generating): Given $t_0 = 11, t_{n+1} = t_n + 4$

  • Sequence: $11, 15, 19, 23, 27, 31, 35\dots$

  • Behavior: Increasing.

  • Value of $t_6$:

    • $t_0=11, t_1=15, t_2=19, t_3=23, t_4=27, t_5=31, t_6=35$.

Writing Recurrence Relations (Chapter 3B)

• Components of a Recurrence Relation:

  1. A starting point (initial value).

  2. A rule that generates successive terms.

• Iterations: Each application of the rule is called an iteration. For $t_n$, $n$ represents the number of times the rule has been applied. Note that $t_0$ has zero iterations.

• General Symbols:

  • $t_0$: Initial or starting term.

  • $t$: Can be replaced by any pronumeral (letter).

• Example 5 (Recurrence Relation Application):

  • Given: $t_0 = 30, I_{n+1} = I_n - 5$

  • The first five terms are:

    • $t_0 = 30$

    • $t_1 = 30 - 5 = 25$

    • $t_2 = 25 - 5 = 20$

    • $t_3 = 20 - 5 = 15$

    • $t_4 = 15 - 5 = 10$

Introduction to Arithmetic Sequences (Chapter 3C)

• Definition: An arithmetic sequence is formed by adding or subtracting a fixed, constant amount to each term.

• Common Difference (D): The fixed amount added/subtracted. It can be found by subtracting any two consecutive terms: $D = t_{n+1} - t_n$.

  • Example: $5, 8, 11, 14, 17, 20\dots$

  • $D = 11 - 8 = 3$ (or $D = 20 - 17 = 3$).

• Identification:

  • Sequence A: $21, 28, 35, 42\dots$ Differences are all 7. This is arithmetic.

  • Sequence B: $2, 6, 18, 54\dots$ Differences are 4, 12, 36. This is not arithmetic (it is geometric).

Tables and Graphs of Arithmetic Sequences

• Linear Relationship: Plotting the term value ($t_n$) against the term number ($n$) results in a straight line.

• Slope: The common difference ($D$) represents the constant slope of the line.

  
  

  • Linear Growth: Occurs if D > 0 (positive slope).

  • Linear Decay: Occurs if D < 0 (negative slope).

  
  

• Example Data:

  
  

  Term Number ($n$)	0	1	2	3	4	5
  Term Value ($t_n$)	5	8	11	14	17	20
  Arithmetic Sequences Using Recursion (Chapter 3D)

  • General Recurrence Form: $t_0 = a, t_{n+1} = t_n + D$

  • Rule for the $n^{th}$ Term: To avoid calculating every intermediate step, use the formula:     $t_n = a + n \times D$

    • Derivation Check:

      • $t_0 = a + 0 \times D$

      • $t_1 = a + 1 \times D$

      • $t_2 = a + 2 \times D$

  • Example Calculation: Find $t_{20}$ for a sequence with $t_0 = 21$ and $D = -3$.

    • $t_{20} = 21 + 20 \times (-3)$

    • $t_{20} = 21 - 60 = -39$

  Finance Applications: Linear Growth and Decay (Chapter 3E)

  Simple Interest

  Simple interest is a recurring arithmetic sequence where the same amount of interest is added each period.

  • Variables:

    • $V_0 = P$: Principal (initial amount).

    • $r$: Interest rate per annum (expressed as a decimal via $\frac{r}{100}$).

    • $D = \frac{r}{100} \times V_0$: Fixed interest added each year.

  • Recurrence Relation: $V_0 = P, V_{n+1} = V_n + D$

  • Partial Periods: If interest is paid for 4 months (out of 12), the amount is $\frac{1}{3}$ of the yearly total.

    • Example: Rate $4.2\%$, Principal $\$3500$.

    • $D = \frac{4.2}{100} \times 3500 = \$147$ per year.

    • Monthly proportion for 4 months: $\frac{147}{3} = \$49$.

  Linear Growth Modeling (Example 13)

  Investment of $\$2000$ at $7.5\%$ simple interest.

  • $D = \frac{7.5}{100} \times 2000 = 150$

  • Recurrence: $V_0 = 2000, V_{n+1} = V_n + 150$

  • Values:

    • $V_1 = 2000 + 150 = \$2150$

    • $V_2 = 2150 + 150 = \$2300$

    • $V_3 = 2300 + 150 = \$2450$

  • Milestone: The investment reaches $\$2750$ after 5 years.

  Depreciation (Linear Decay)

  • Flat Rate/Fixed Rate Depreciation: The value decreases by a fixed amount ($D$) each year. The process is identical to simple interest, but $D$ is negative (D < 0).

  General Rule for Linear nth Term

  $V_n = V_0 + n \times D$

  • Example 16: Investment of $\$4000$ at $6.5\%$ interest ($D = \$260$).

    • Value after 15 years: $V_{15} = 4000 + 15 \times 260 = \$7900$.

    • When will it exceed $\$10,000$?

      • $10000 = 4000 + n \times 260$

      • $6000 = n \times 260$

      • $n = \frac{6000}{260} \approx 23.07$ years.

      • The value first exceeds $\$10,000$ after 24 years.

  Introduction to Geometric Sequences (Chapter 3F)

  • Definition: A sequence formed by multiplying the previous term by a fixed number called the Common Ratio (R).

  • Finding R: Divide any term by the term immediately preceding it:     $R = \frac{t_{n+1}}{t_n} = \frac{t_1}{t_0} = \frac{t_2}{t_1} \dots$

  • Example: $5, 10, 20, 40, 80, 160\dots$

    • $R = \frac{10}{5} = \frac{20}{10} = 2$

  Graphs and Behaviors of Geometric Sequences

  • Geometric Growth (R > 1):

    • Graph: A curve with an increasing positive slope.

    • Example: $2, 6, 18, 54 \dots$ ($R = 3$).

  • Geometric Decay (0 < R < 1):

    • Graph: A curve with a decreasing negative slope.

    • Example: $32, 16, 8, 4 \dots$ ($R = 0.5$).

  Converting Percentage Change to R

  To use geometric sequences in finance, percentage changes must be converted to a common ratio ($R$). Think of $1$ as the status quo ($100\%$).

  • Percentage Increase: $R = 1 + \frac{r}{100}$

    • $20\%$ increase: $R = 1 + 0.2 = 1.2$

    • $10\%$ increase: $R = 1.1$

    • $5\%$ increase: $R = 1.05$

    • $3.8\%$ increase: $R = 1.038$

  • Percentage Decrease: $R = 1 - \frac{r}{100}$

    • $20\%$ decrease: $R = 1 - 0.2 = 0.8$

    • $10\%$ decrease: $R = 0.9$

    • $5\%$ decrease: $R = 0.95$

    • $3.8\%$ decrease: $R = 0.962$

  Geometric Sequence Recursion (Chapter 3G)

  • General Recurrence Form: $t_0 = a, t_{n+1} = R \times t_n$

  • Example 21: Generate first five terms for $t_0 = 5, t_{n+1} = 2t_n$

    • $t_0 = 5$

    • $t_1 = 10$

    • $t_2 = 20$

    • $t_3 = 40$

    • $t_4 = 80$

  • Rule for the $n^{th}$ Term:     $t_n = R^n \times a$

    • Example: Find $t_{12}$ for $t_0 = 5, R = 2$ .

    • $t_{12} = 2^{12} \times 5 = 4096 \times 5 = 20480$.

  Finance Applications: Geometric Growth and Decay (Chapter 3H/3I)

  Compound Interest

  Compound interest occurs when interest earned is added to the principal, allowing the total to earn more interest in subsequent periods.

  • Variables:

    • $V_0$: Principal.

    • $r$: Annual interest rate in percent.

    • $R = 1 + \frac{r}{100}$

  • Recurrence: $V_0 = \text{principal}, V_{n+1} = R \times V_n$

  • Rule (n periods): $V_n = R^n \times V_0$

  Reducing-Balance Depreciation

  This is geometric decay where an asset loses value by a fixed percentage each year.

  • Recurrence: $V_0 = \text{principal}, V_{n+1} = R \times V_n$

  • Ratio: $R = 1 - \frac{r}{100}$

  • Rule (n periods): $V_n = R^n \times V_0$

  Integrated Examples (Example 26)

  • Investment Growth: $\$1000$ investment at $10\%$ compounded per year for 15 years.

    • $V_0 = 1000, R = 1.1$

    • $V_{15} = 1.1^{15} \times 1000 \approx \$4177.24$

    • To the nearest dollar: $\$4177$.

  • Car Depreciation: $\$18500$ car depreciating at $10\%$ per year for 12 years.

    • $V_0 = 18500, R = 0.9$

    • $V_{12} = 0.9^{12} \times 18500 \approx \$5224.94$

    • To the nearest dollar: $\$5225$.