Patterns, Sequences, and Series Study Guide

Fundamental Concepts of Patterns and Sequences

Pattern Definition: A regular and repeatedly occurring arrangement of numbers, shapes, colors, or objects. Patterns operate based on specific rules that enable the prediction of subsequent elements.
Visual Pattern: A series comprised of shapes, images, or objects organized in a predictable and repeated manner. These are frequently observed in the natural world and various forms of art.
Number Sequence: An ordered list of numbers governed by a specific rule.
- Term: The designation for each individual number within a sequence.
Fibonacci Sequence: A distinct sequence where each term results from the sum of the two immediately preceding terms. It typically commences with $0$ and $1$ (or $1$ and $1$ in certain variations).
Pattern Rule: The specific method or mathematical operation applied to progress from one term to the next. This rule can be articulated through words, mathematical formulas, or equations.

Classifications of Patterns

Repeating Pattern: A pattern wherein the exact same sequence of elements repeats indefinitely.
- Example: A sequence of colors such as red-blue, red-blue, red-blue.
Growing Pattern: A pattern that undergoes an increase or change according to a defined rule.
- Example: The numerical sequence $2, 4, 6, 8, \dots$
Symmetrical Pattern: A design where one half serves as a mirror image of the other half.
- Examples: The wings of a butterfly or a design created by folding paper.
Geometrical or Visual Pattern: Specific arrangements of shapes, figures, colors, or designs organized in a structured format.
- Example: Tiles on a floor or decorative designs in artwork.

Patterns in Nature and Arts

Nature

Nature demonstrates mathematical patterns through various structures:

Spiral Arrangements: These patterns curve around a central focal point. Common examples include shells, flowers, and specific plant types.
Branching Patterns: Structures that divide into progressively smaller parts. These are visible in tree branches, root systems, and river networks.
Symmetry: Two halves that mirror or closely resemble each other, found in butterflies, leaves, and flowers.
Growth Patterns: The systematic ways living organisms develop over time, such as leaf arrangements. Many follow structured numerical relationships like the Fibonacci pattern.

Arts

Artists utilize mathematical concepts to achieve aesthetic goals:

Repetition: The repeated application of elements (lines, colors, shapes) to establish rhythm and unity.
Tessellation: A pattern created by repeating shapes that interlock perfectly without any gaps or overlapping edges.
Symmetry: A balanced composition where one side of the artwork mirrors the opposite side.
Proportion: The relationship between the sizes of various components within a work, used to create harmony.
Geometric Arrangements: Organizing squares, circles, triangles, and polygons into structured visual patterns, which can often be converted into numerical relationships.

Activity: Predicting Terms and Determining Rules

Sequence: $2, 3, 5, 8, 12, 17, 23, \dots$
- Solution:
  - $2 + 1 = 3$
  - $3 + 2 = 5$
  - $5 + 3 = 8$
  - $8 + 4 = 12$
  - $12 + 5 = 17$
  - $17 + 6 = 23$
  - $23 + 7 = 30$
- Next Term: $30$
- Rule: Each term is obtained by adding the next consecutive whole number ( $1, 2, 3, 4, \dots$ ) to the preceding term.
Sequence: $2, 4, 8, 16, 32, \dots$
- Solution:
  - $2 \times 2 = 4$
  - $4 \times 2 = 8$
  - $8 \times 2 = 16$
  - $16 \times 2 = 32$
  - $32 \times 2 = 64$
- Next Term: $64$
- Rule: Multiply the previous term by $2$ to generate the subsequent term.
Sequence: $100, 90, 81, 73, 66, \dots$
- Solution:
  - $100 - 10 = 90$
  - $90 - 9 = 81$
  - $81 - 8 = 73$
  - $73 - 7 = 66$
  - $66 - 6 = 60$
- Next Term: $60$
- Rule: Each term is obtained by subtracting consecutive whole numbers in decreasing order ( $10, 9, 8, 7, 6, \dots$ ) from the previous term.
Sequence: $256, 128, 64, 32, 16, \dots$
- Solution:
  - $256 \div 2 = 128$
  - $128 \div 2 = 64$
  - $64 \div 2 = 32$
  - $32 \div 2 = 16$
  - $16 \div 2 = 8$
- Next Term: $8$
- Rule: Divide the previous term by $2$ to get the next term.

Fibonacci Sequence in Depth

Definition and Core Rule

In a Fibonacci sequence, each term is the sum of the two preceding terms.

Standard Sequence: $0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \dots$
Mathematical Formula: $F_n = F_{n-1} + F_{n-2}$

Illustrative Examples

Example 1: Rabbit Population Model: A pair of newborn rabbits (one male, one female) is placed in a field. They take one month to mature. After maturing, they mate and produce one new male-female pair every month. Assuming no rabbits die:
- January: 1 pair (juvenile)
- February: 1 pair (matured adult)
- March: 2 pairs (original adult pair + 1 new juvenile pair)
- Continuance of this logic follows the Fibonacci sequence to determine the total pairs after one year.
Example 2: Sunflower Spiral Growth: A sunflower develops new seed spirals weekly.
- Week 1: $1$ spiral
- Week 2: $1$ spiral
- Subsequent weeks: The sum of spirals from the previous two weeks.
- Calculation for 8 weeks: The sequence is $1, 1, 2, 3, 5, 8, 13, 21$ .
- Result: After 8 weeks, the sunflower will have $21$ spirals.

Arithmetic Sequences

Definitions

Sequence: A function with a domain consisting of the set of positive integers; an ordered list of numbers.
Arithmetic Sequence: A sequence where every term after the first is determined by adding a constant value.
Common Difference ( $d$ ): The constant value added to terms in an arithmetic sequence. It is found using $d = a_n - a_{n-1}$ .

Explicit Formula

$a_n = a_1 + (n - 1)d$

$a_n$ : The $n^{th}$ term
$a_1$ : The first term
$n$ : The number of terms
$d$ : The common difference

Attributes and Real-Life Interpretation

Constant Difference: The gap between consecutive terms remains uniform (e.g., in $5, 9, 13, 17, 21$ , the difference is always $4$ ).
Linear Growth: Arithmetic sequences increase or decrease at a steady, constant rate. This creates a straight-line pattern when graphed.
Applications:
- Annual salary increases (e.g., rising by $Php2,000$ every year).
- Consistent weekly savings (e.g., $Php500$ per week).
- Distance traveled when moving at a constant speed.

Arithmetic Sequence Examples

Alex’s Salary (Company A): Starting monthly salary of $Php15,000$ with an annual increase of $Php1,000$ .
- To find the salary in the $5^{th}$ year ( $n=5$ ):
- $a_5 = 15000 + (5-1)1000 = 15000 + 4000 = 19000$
- Result: $Php19,000$
Davao Scholarship Program:
- Batch 1: $1,040$ scholars
- Batch 2: $1,115$ scholars
- Batch 3: $1,190$ scholars
- Common difference $d = 1115 - 1040 = 75$
- To find scholars for the $23^{rd}$ batch ( $n=23$ ):
- $a_{23} = 1040 + (23-1)75 = 1040 + (22)75 = 1040 + 1650 = 2690$
- Result: $2,690$ scholars

Geometric Sequences

Definitions

Geometric Sequence: A sequence where each term is calculated by multiplying the previous term by a constant multiplier.
Common Ratio ( $r$ ): The constant multiplier, found using $r = \frac{a_n}{a_{n-1}}$ .

Explicit Formula

$a_n = a_1 \times r^{n-1}$

$a_n$ : The $n^{th}$ term
$a_1$ : The first term
$r$ : The common ratio
$n$ : The term number

Attributes and Growth Dynamics

Constant Ratio: The quotient of any term divided by its predecessor is identical (e.g., in $2, 6, 18, 54$ , the ratio is $3$ ).
Exponential Growth/Decay: Use of repeated multiplication causes values to change by a percentage or factor rather than a fixed amount. Powers increase with each step.
Real-Life Growth Examples:
- Population expansion.
- Compound interest in finance.
- Viral spread of information or diseases.
- Investment growth.
Real-Life Decay Examples:
- Asset depreciation.
- Radioactive decay.
- Decreasing populations.

Geometric Sequence Examples

Alex’s Salary (Company B): Starting monthly salary of $Php15,000$ with a $7\%$ annual increase.
- $r = 1 + 0.07 = 1.07$ ; $n = 5$
- $a_5 = 15000 \times (1.07)^{5-1} = 15000 \times (1.07)^4$
- Result: $Php19,661.94$
Cookie Production: Day 1 produces $5$ boxes; each day increases by $3$ times the previous day.
- $a_1 = 5$ ; $r = 3$ ; $n = 8$
- $a_8 = 5\times(3)^{8-1} = 5\times(3)^7 = 5\times(2187) = 10935$
- Result: $10,935$ boxes produced on the $8^{th}$ day.

Sequence Practice Problems

PDF Zoom Problem: A PDF magnification increases by $15\%$ per zoom level. Original length of "January" is $1.2\,cm$ . Find length after $6$ magnifications.
- Type: Geometric
- Given: $a_1 = 1.2$ , $r = 1.15$ , $n = 6$
- Solution: $a_6 = 1.2(1.15)^{6-1} = 1.2(1.15)^5 \approx 2.41\,cm$
- Result: $2.41\,cm$
Content Creator Followers: $20$ followers on Day 1. Each day increases by $50\%$ more than the previous.
- Type: Geometric
- Given: $a_1 = 20$ , $r = 1.50$ , $n = 7$
- Solution: $a_7 = 20(1.50)^{7-1} = 20(1.50)^6 \approx 227.81$
- Result: $228$ followers on the $7^{th}$ day.
Tree-Planting Activity: Month 5 planted $180$ trees; Month 12 planted $355$ trees. Increase is constant.
- Type: Arithmetic
- Given: $a_5 = 180$ , $a_{12} = 355$
- Solution:
  - Change in trees: $355 - 180 = 175$
  - Change in months: $12 - 5 = 7$
  - Common difference $d = \frac{175}{7} = 25$
  - Find $a_1$ : $180 = a_1 + 25(5-1) \rightarrow 180 = a_1 + 100$
  - $a_1 = 80$
- Result: $80$ trees were planted in the first month.

Arithmetic and Geometric Series

Definitions

Series: The total sum of the terms within a specified sequence.

Arithmetic Series Formula

There are two primary formulas based on the provided data:

If the first and last terms are known: $S_n = \frac{n(a_1 + a_n)}{2}$
If the first term, difference, and number of terms are known: $S_n = \frac{n}{2}[2a_1 + (n - 1)d]$

Geometric Series Formula

Used to sum terms in a geometric sequence:

$S_n = \frac{a_1(1 - r^n)}{1 - r}$ , provided that $r \neq 1$

Series Applications and Examples

Arithmetic Series Examples

Alex’s Total Career Earnings (Company A): 10-year total. Salary is fixed for the year before increasing.
- Year 1 Annual: $15000 \times 12 = 180,000$
- Year 2 Annual: $16000 \times 12 = 192,000$
- $d = 12,000$ ; $n = 10$ ; $a_1 = 180,000$
- $S_{10} = \frac{10}{2}[2(180000) + (10-1)12000] = 5[360000 + 108000] = 5[468000]$
- Result: $Php2,340,000$
Sum of First 100 Odd Numbers:
- $a_1 = 1$ , $d = 2$ , $n = 100$
- $S_{100} = \frac{100}{2}[2(1) + (100-1)2] = 50[2 + 198] = 50[200]$
- Result: $10,000$

Geometric Series Examples

Alex’s Total Career Earnings (Company B): 10-year total based on $7\%$ annual increase.
- $a_1 = 180,000$ ; $r = 1.07$ ; $n = 10$
- $S_{10} = \frac{180000(1 - 1.07^{10})}{1 - 1.07}$
- Result: $Php2,486,960.63$
Vince’s Hiking Trip: 3-day trip. Each day distance is $3/4$ (or $0.75$ ) of the previous day. Total distance is $83.25\,km$ .
- $S_3 = 83.25$ ; $r = 0.75$ ; $n = 3$
- $83.25 = \frac{a_1(1 - 0.75^3)}{1 - 0.75}$
- After solving for $a_1$
- Result: $36\,km$ (the distance on the first day).

Series Practice Problems

Theater Seating: $32$ rows total. Row 1 has $26$ seats, Row 2 has $30$ , Row 3 has $34$ , etc.
- Type: Arithmetic Series
- Given: $a_1 = 26$ , $d = 4$ , $n = 32$
- Solution: $S_{32} = \frac{32}{2}[2(26) + (32-1)4] = 16[52 + 124] = 16[176] = 2816$
- Result: $2,816$ seats total.
Jerry’s Investment: Deposit of $Php20,000$ with a yearly return of $Php1,750$ . Total after $8$ years.
- Type: Arithmetic Series
- Given: $a_1 = 20,000$ , $d = 1,750$ , $n = 8$
- Solution: $S_8 = \frac{8}{2}[2(20000) + (8-1)1750] = 4[40000 + 12250] = 4[52250] = 209000$
- Result: $Php209,000$
Deer Population: Initial population of $2,537$ ; growth factor is $1.025$ per year. Find total deer counted over $10$ years.
- Type: Geometric Series
- Given: $a_1 = 2,537$ , $r = 1.025$ , $n = 10$
- Solution: $S_{10} = \frac{2537(1 - 1.025^{10})}{1 - 1.025} \approx 28423$
- Result: $28,423$ deer.
Shopping Mall Traffic: $120$ shoppers on Day 1. Each subsequent day traffic increases by $10\%$ .
- Type: Geometric Series
- Given: $a_1 = 120$ , $r = 1.10$ , $n = 7$
- Solution: $S_7 = \frac{120(1 - 1.10^7)}{1 - 1.10} \approx 1138.45$
- Result: $1,139$ shoppers (rounded to nearest integer).