Core Sequences and Arithmetic Series Study Guide

Essential Understandings and Core Concepts

Number and algebra allow the representation of patterns, the demonstration of equivalences, and the formulation of generalizations that enable the modeling of real-world situations.
Algebra serves as an abstraction of numerical concepts, employing variables to systematically solve mathematical problems.
Key learning outcomes include:
- Utilizing the formula for the $n\text{th}$ term of an arithmetic sequence.
- Utilizing the formula for the sum of the first $n$ terms of an arithmetic sequence.
- Utilizing the formula for the $n\text{th}$ term of a geometric sequence.
- Utilizing the formula for the sum of the first $n$ terms of a geometric sequence.
- Working with sigma notation ( $\sum$ ) for sums of arithmetic and geometric sequences.
- Applying arithmetic and geometric sequences to solve real-world problems.
Core concepts addressed:
- Modelling real-life situations using the structure of arithmetic and geometric sequences and series enables prediction, analysis, and interpretation.
- Formulae represent generalizations constructed on the basis of specific examples, which can subsequently be extended to new examples.
- Mathematical financial models, such as compounded growth, allow the computation, evaluation, and interpretation of debt and investment both approximately and accurately.

Prior Knowledge Assessment

System of simultaneous linear equations to solve:

$2x + 5y = 3$

$3x - y = 13$

Numerical evaluation tasks:
- Evaluate $5 \times 2^4$
- Evaluate $4 \times (-3)^2$
Polynomial equation solution task:
- Find all solutions to $x^4 = 16$

Patterns Across Mathematics, Science, and Art

Patterns serve as a primary area of overlap between mathematics, science, and art:
- In art, patterns may appear purely aesthetic, but artists throughout history have incorporated mathematical concepts into drawings and sculptures—ranging from the ancient Greeks to Leonardo da Vinci and modern graphic artist M. C. Escher.
- In science, recurring patterns observed in experimental data lead directly to advances in theoretical understanding.
- In mathematics, establishing an understanding of simple structural relationships facilitates the modeling of complex patterns.
Biological example of exponential doubling patterns in cell development:
- The early development of a fertilized egg demonstrates a pattern of rapid cell division:
  - Zygote ( $1$ cell stage, $2^0$ )
  - $2$ Cell Stage ( $2^1$ )
  - $4$ Cell Stage ( $2^2$ )
  - $8$ Cell Stage ( $2^3$ )
  - Morula ( $72\,\text{hours}$ )
  - Blastocyst ( $4\,\text{days}$ )

Early development of a fertilized egg - Zygote, 2 Cell Stage, 4 Cell Stage

Early development of a fertilized egg - 8 Cell Stage, Morula, Blastocyst

Starter Problem (Financial Choice Analysis):
- In early $2018$ , an $18$ -year-old Canadian woman won a lottery jackpot after buying a scratch card for the first time.
- She was presented with two tax-free options:
  - Option $1$ : A lump sum payment of $C\$1\,000\,000$
  - Option $2$ : An annual payout of $C\$50\,000$ per year for life

Arithmetic Sequences

An arithmetic sequence is formed by adding or subtracting a constant value to obtain each successive term.
This constant added or subtracted value is designated as the common difference, $d$ (which can be positive or negative).
Examples of arithmetic sequences:
- $1, 4, 7, 10, 13, \dots$ has a common difference of $d = +3$
- $19, 15, 11, 7, 3, \dots$ has a common difference of $d = -4$
Standard notation for sequence terms:
- First term: $u_1$
- Second term: $u_2 = u_1 + d$
- Third term: $u_3 = u_1 + 2d$
- Fourth term: $u_4 = u_1 + 3d$
- Fifth term: $u_5 = u_1 + 4d$
- $n\text{th}$ term: $u_n$
Formula for the $n\text{th}$ term of an arithmetic sequence:

$u_n = u_1 + (n - 1)d$

Generalization concept note:
- Formally proving this formula requires mathematical induction.
- Systematic listing allows generalization of the pattern.
- An alternative expanded form of the equation is $u_n = u_1 + nd - d$
Worked Example 2.1:
- Problem: An arithmetic sequence has first term $u_1 = 7$ and common difference $d = 8$ . Find the $10\text{th}$ term ( $u_{10}$ ).
- Solution step 1: Write the general formula with known values:

$u_n = 7 + 8(n - 1)$

*   Solution step 2: Substitute  $n = 10$ :

$u_{10} = 7 + 8(10 - 1) = 7 + 8(9) = 79$

Worked Example 2.2:
- Problem: The third term of an arithmetic sequence is $10$ ( $u_3 = 10$ ) and the $15\text{th}$ term is $46$ ( $u_{15} = 46$ ). Find the first term ( $u_1$ ) and the common difference ( $d$ ).
- Solution step 1: Express given terms using the $n\text{th}$ term formula:

$u_3 = 10 \implies u_1 + 2d = 10$

$u_{15} = 46 \implies u_1 + 14d = 46$

*   Solution step 2: Solve the system of linear equations simultaneously using elimination or Graphic Display Calculator (GDC) linear system solver:

$(u_1 + 14d) - (u_1 + 2d) = 46 - 10$

$12d = 36 \implies d = 3$

$u_1 + 2(3) = 10 \implies u_1 = 4$

*   Result:  $u_1 = 4$ ,  $d = 3$

Worked Example 2.3:
- Problem: Find the total number of terms $n$ in the arithmetic sequence $1, 4, 7, \dots, 100$
- Solution step 1: Identify initial conditions:

$u_1 = 1$

$d = 3$

*   Solution step 2: Formulate the  $n\text{th}$  term equation:

$u_n = 1 + 3(n - 1)$

*   Solution step 3: Set  $u_n = 100$  and solve for  $n$ :

$1 + 3(n - 1) = 100$

$3(n - 1) = 99$

$n - 1 = 33$

$n = 34$

Sums of Arithmetic Sequences (Arithmetic Series)

An arithmetic series is the sum of the terms of an arithmetic sequence.
Notation conventions:
- First term: $u_1$
- Sum of first two terms: $S_2 = u_1 + u_2$
- Sum of first three terms: $S_3 = u_1 + u_2 + u_3$
- Sum of first $n$ terms: $S_n$
Formulas for the sum of the first $n$ terms of an arithmetic sequence:
- Version 1 (using first term $u_1$ and common difference $d$ ):

$S_n = \frac{n}{2}[2u_1 + (n - 1)d]$

*   Version 2 (using first term  $u_1$  and last term  $u_n$ ):

$S_n = \frac{n}{2}(u_1 + u_n)$

Worked Example 2.5:
- Problem: An arithmetic sequence has first term $u_1 = 10$ and last term $u_n = 1000$ . If there are $n = 30$ terms, find the sum of all the terms ( $S_{30}$ ).
- Solution step 1: Apply the sum formula incorporating the first and last terms:

$S_n = \frac{n}{2}(u_1 + u_n)$

*   Solution step 2: Substitute  $u_1 = 10$ ,  $u_n = 1000$ , and  $n = 30$ :

$S_{30} = \frac{30}{2}(10 + 1000) = 15(1010) = 15150$

Sigma Notation for Arithmetic Series

Sigma notation provides a concise shorthand method for expressing the sum of sequence terms using the capital Greek letter sigma ( $\sum$ ).
General definition of sigma notation:

$\sum_{r=1}^{n} u_r = u_1 + u_2 + u_3 + \dots + u_n$

*   The lower boundary (e.g.,  $r = 1$ ) specifies the starting point of summation.

*   The upper boundary (e.g.,  $r = n$ ) specifies the stopping point of summation.

*   The dummy variable  $r$  can be substituted with any variable symbol.

Critical Counting Caveat:
- When evaluating sums where the lower index is greater than $1$ , count terms carefully.
- The formula for the total number of terms between lower limit $a$ and upper limit $b$ is n = b - a + 1$.\n\n * Example: In the summation \sum_{r=3}^{10} (5r + 2) $, there are$ 10 - 3 + 1 = 8 $terms ($ u_3, u_4, u_5, u_6, u_7, u_8, u_9, u_{10} $), rather than$ 7 terms.\n\n* Worked Example 2.6:\n\n * Problem: Evaluate the summation:\n\n \sum_{r=3}^{10} (5r + 2)\n\n * Solution step 1: Substitute initial values of r to establish the sequence properties:\n\n For r = 3 $:$ 5(3) + 2 = 17\n\n For r = 4 $:$ 5(4) + 2 = 22\n\n For r = 5 $:$ 5(5) + 2 = 27\n\n * Solution step 2: Identify sequence parameter values:\n\n u_1 = 17\n\n d = 5\n\n n = 10 - 3 + 1 = 8\n\n * Solution step 3: Apply the series sum formula S_n = \frac{n}{2}[2u_1 + (n - 1)d]:\n\n S_8 = \frac{8}{2}[2(17) + (8 - 1)5] = 4[34 + 35] = 4(69) = 276$$
- Calculator implementation: Sums given in sigma notation can also be evaluated directly using GDC summation commands (e.g., \Sigma(5R+2, R, 3, 10) = 276).