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Comprehensive Guide to Patterns and Sequences in Mathematics

Explore patterns in nature and art, understand arithmetic and geometric sequences, learn the Fibonacci sequence, and solve real-world problems using mathematical sequences.

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Patterns and Sequences in General Mathematics
Learning Objectives Describe patterns by inspection, including in art and nature Determine the next term in various patterns including Fibonacci sequence Identify the rule governing a pattern and illustrate attributes of arithmetic and geometric sequences
Patterns in Art and Nature Discover how patterns appear all around us in nature and art
What is a Pattern? A pattern is a repeated or predictable arrangement of numbers, shapes, or objects. Patterns follow specific rules that allow us to predict what comes next. They can be visual (like shapes and colors) or numerical (like number sequences). Examples: 2, 4, 6, 8... or ... ▲ ● ▲ ● ▲ ●
Recognizing Patterns by Inspection Look for repetition or regular changes in sequences Notice increases or decreases in numbers Identify recurring shapes or colors. Example: 2, 4, 6, 8… (increasing by 2)
Determining the Next Term in Patterns To find the next term, identify the rule connecting each term. Look at the pattern: 3, 6, 12, 24, ? Each term is double the previous term (×2). Following this rule, the next term is 24 × 2 = 48! Try it yourself: What comes next in 5, 10, 15, 20, ?
The Fibonacci Sequence The famous Fibonacci sequence appears in many natural forms
Understanding the Fibonacci Rule The Fibonacci sequence follows a simple but powerful rule: each term is the sum of the two previous terms. Starting with 0 and 1, we get: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... Interactive Challenge: What is the 10th Fibonacci number?
Identifying Pattern Rules Addition or subtraction (arithmetic patterns) Multiplication or division (geometric patterns) Alternating, repeating, or Fibonacci (recursive) sequences Example: Rule for 2, 4, 8, 16 is "multiply by 2"
Arithmetic Sequences: Definition & Attributes An arithmetic sequence is a sequence where each term is found by adding a constant value (called the common difference) to the previous term. Key attributes include the first term (a₁) and common difference (d). Example: 3, 7, 11, 15, … where d = 4
Attributes of Arithmetic Sequences Illustrated General term formula: aₙ = a₁ + (n - 1)d Constant difference between consecutive terms Linear growth or decline pattern. Example: For 3, 7, 11, 15... find the 10th term: a₁₀ = 3 + (10-1)×4 = 39
Classifications of Sequence FINITE SEQUENCE A finite sequence has a specific number of terms – it has a beginning and an end. Example: 2, 4, 6, 8, 10 (This sequence has 5 terms – it ends at 10)
Classifications of Sequence INFINITE SEQUENCE An infinite sequence has no end- it goes on forever. It keeps following the same rule endlessly. You can describe the rule, but you can’t list all the terms. Example: 1, 2, 4, 8, 16, … (This sequence keeps doubling and never stops)
Classifications of Sequence “…” In mathematical notation, an infinite sequence often includes ellipsis “…” to show it continues.
Problem: Find the 20th term of 5, 8, 11, 14... Step 1: Identify a₁ = 5, d = 3 Step 2: Apply formula a₂₀ = 5 + (20- 1)×3 = 62 Step 12: Arithmetic Sequence Problem
Geometric Sequences – Definition and Attributes A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant ratio. Key attributes include the common ratio (r) and the first term (a₁). Example: 2, 6, 18, 54, … (r = 3)
Attributes of Geometric Sequences Illustrated General term formula: aₙ = a₁ × r(n-1) Constant ratio between consecutive terms Exponential growth (r > 1) or decay (0 < r < 1) Example: For 2, 6, 18, 54... find the 5th term: a₅ = 2 × 3⁴ = 162
Find the 8th term: 3, 6, 12, 24, ... a₁ = 3, r = 2 a₈ = 3 × 2⁷ = 384 → Step 15: Geometric Sequence Problem
Comparing Arithmetic and Geometric Sequences Arithmetic: Add constant difference → Linear growth (e.g., 2, 5, 8, 11...) Geometric: Multiply constant ratio → Exponential growth (e.g., 2, 6, 18, 54...) Key difference: Arithmetic grows steadily, Geometric grows rapidly!
Solving Real- World Problems with Sequences Arithmetic Example: You save $50 more each month. Starting with $100, your savings grow: $100, $150, $200, $250... Use aₙ = a₁ + (n-1)d to find any month's total! Geometric Example: Bacteria double every hour: 1, 2, 4, 8, 16...
Key Takeaways Patterns appear everywhere in nature and art Fibonacci is a special recursive pattern Arithmetic adds constant; geometric multiplies ratio
ARITHMETIC SEQUENCE | EXAMPLES Example 1 The sequence is: 7, 12, 17, 22, ... Find the 20th term (a20) Given: a₁ = 7; n = 20 d = a2 – a1 d = 12 – 7 d = 5 Formula: aₙ = a1 + (n - 1)d Solution: a20 = a1 + (n - 1)d a20 = 7 + (20– 1) x 5 a20 = 7 + (19) x 5 a20 = 7 + 95 a20 = 102
ARITHMETIC SEQUENCE | EXAMPLES Example 2 Find the 30th term of the arithmetic sequence: −15, −9, −3, 3, ... Find: a30 Given: a₁ = -15; n = 30 d = a2 – a1 d = -9 – -15 d = 6 Formula: aₙ = a1 + (n - 1)d Solution: a30 = a1 + (n - 1)d a30 = -15 + (30– 1) x 6 a30 = -15 + (29) x 6 a30 = -15 + 174 a30 = 159
ARITHMETIC SEQUENCE | EXAMPLES Example 3 The second term of an arithmetic sequence is 18, and the common difference is −4. Find the 100th term (a100) Given: d = - 4; n = 35 a₁ = a2 – d a₁ = 18 – (-4) a₁ = 22 Formula: aₙ = a1 + (n - 1)d Solution: a100 = a1 + (n - 1)d a100 = 22 + (100 – 1) x -4 a100 = 22 + (99) x -4 a100 = 22 - 396 a100 = -374
ARITHMETIC SEQUENCE | EXAMPLES Solution: a18 = a1 + (n - 1)d a18 = 500 + (18 – 1) x 150 a18 = 500 + (17) x 150 a18 = 500 + 2550 a18 = ₱3050 Word Problem 1 – Savings Maria saves ₱500 during the first week. She decides to increase the amount she saves by ₱150 every week. If this pattern continues, how much will Maria save during the 18th week? Find the 18th week savings (a18) Given: a₁ = ₱500 ; n = 18th week d = ₱150 Formula: aₙ = a1 + (n - 1)d
ARITHMETIC SEQUENCE | EXAMPLES Solution: a24 = a1 + (n - 1)d a24 = 25000 + (24 – 1) x 800 a24 = 25000 + (23) x 800 a24 = 25000 + 18400 a24 = ₱43400 Word Problem 2 – Salary Increase A newly hired employee receives a monthly salary of ₱25,000 during the first month. Every succeeding month, his salary increases by ₱800. What will be his salary during the 24th month? Find the 24th month salary increase (a24) Given: a₁ = ₱25,000 ; n = 24th week d = ₱800 Formula: aₙ = a1 + (n - 1)d
GEOMETRIC SEQUENCE | EXAMPLES Example 1 Find the 12th term of the sequence: 3, 6, 12, 24, ... Find the 12th term (a12) Given: a₁ = 3 r = a2 / a1 r = 6 / 3 r = 2 n = 12 Formula: aₙ = a₁ × r(n-1) Solution: a12 = a₁ × r (n-1) a12 = 3 × 2 (12-1) a12 = 3 × 2 (11) a12 = 3 × 2048 a100 = 6144
GEOMETRIC SEQUENCE | EXAMPLES Example 2 Determine the 15th term of the sequence: 2, 8, 32, 128, ... Find the 15th term (a15) Given: a₁ = 2 r = a2 / a1 r = 8 / 2 r = 4 n = 15 Formula: aₙ = a₁ × r(n-1) Solution: a15 = a₁ × r (n-1) a15 = 2 × 4 (15-1) a15 = 2 × 4 (14) a15 = 2 × 268,435,456 a15 = 536,870,912
GEOMETRIC SEQUENCE | EXAMPLES Example 3 The third term of a geometric sequence is 96, and the common ratio is ½. Find the 10th term. Find the 10th term (a10) Given: r = n = 10; a3 = 96 a2 = a3 / r a2 =96 / 0.5 = 192 a1 = 192 / 0.5 = 384 Formula: aₙ = a₁ × r(n-1) Solution: a10 = a₁ × r (n-1) a10 = 384 × (10-1) a10 = 384 × (9) a10 = 384 × a10 = 0.75
GEOMETRIC SEQUENCE | EXAMPLES Solution: a8 = 10000 × 1.20 (8-1) a8 = 10000 × 1.20 (7) a8 = 10000 × 3.5832 a8 = ₱35,831.81 Word Problem 1 – Investment Carlos invested ₱10,000 in a business that grows by 20% every year. Assuming the investment grows according to a geometric sequence, what will be its value at the 8th year? Find the 8th year investment value (a18) Given: a₁ = ₱10,000 ; n = 8th year r = 100%(principal amount) + 20% r = 120% or 1.20 Formula: aₙ = a₁ × r(n-1)
GEOMETRIC SEQUENCE | EXAMPLES Solution: a9 = 200 × 2 (9-1) a9 = 200 × 2 (8) a9 = 200 × 256 a9 = 51,200 bacteria Word Problem 2 – Bacteria Growth A laboratory culture contains 200 bacteria. Every hour, the number of bacteria doubles. How many bacteria are present after the 9th hour? Find: No. of bacteria present after the 9th hour (a9) Given: a₁ = 200 bacteria ; n = 9th hour r = 2 (the no. of bacteria doubles every hour Formula: aₙ = a₁ × r(n-1)
ASSIGNMENT 1. A patient takes 80 mg of medicine on the first day. Because of recovery, the dosage is reduced by half every day. What will be the dosage on the 7th day? 2. A community plants 45 trees during the first month of a reforestation project. Each month, they plant 12 more trees than they did the previous month. If they continue this pattern, how many trees will they plant during the 24th month?