Year 8 Unit 13: Sequences Study Guide

Unit Overview: Year 8 Unit 13 - Sequences

• This unit covers the identification, description, and generation of various numerical and visual sequences.
• Emphasis is placed on using different rules (term-to-term and position-to-term) and identifying specific types of progressions such as linear, geometric, quadratic, and Fibonacci.

Learning Outcomes

Support Level

• Describe how a sequence continues: Identify the next numbers in a list based on an observed pattern.
• Rules in words: Know how to express a rule for a sequence using verbal descriptions.
• Term-to-term rules: Generate sequences when provided with a specific rule to get from one term to the next.
• Diagram patterns: Describe patterns shown by diagrams or visual images.
• Special sequences: Recognize and use sequences of triangular, square, and cube numbers.

Core Level

• Generating from $n^{th}$ term: Know how to generate a sequence when the $n^{th}$ term formula is given.
• Diagram derivation: Generate simple sequences derived from visual patterns or diagrams.
• Linear $n^{th}$ term: Find the $n^{th}$ term formula for a linear (arithmetic) sequence.

Extension Level

• Fibonacci-type sequences: Work with sequences where terms are found by adding previous terms.
• Quadratic sequences: Know how to continue terms of a quadratic sequence.
• Geometric progressions: Work out the value of a term in a geometrical progression of the form $ar^{n-1}$, where $n$ is an integer $> 0$.

Reference Codes and Resources

• Sparx Codes:   - M381   - U530   - M241   - M981   - M991   - U498   - U680   - M418   - U213
• Corbett Maths Videos:   - Reference numbers: 286, 287, 287a, 288, 289, 290, 212, 226, 229, 388, 375.

Key Concepts and Definitions

• Sequence: A list of numbers or items in a specific order.
• $n^{th}$ term: A formula (position-to-term rule) that allows you to calculate the value of any term in a sequence based on its position $n$.
• Arithmetic or Linear Sequences: Sequences that increase or decrease by a common amount (a common difference) each time.
• Geometric Series/Sequence: A sequence that has a common multiple (common ratio) between each term.
• Quadratic Sequences: Sequences that include an $n^2$ term. These sequences have a common second difference.
• Fibonacci Sequences: Sequences where you add the two previous terms to find the next term.

Arithmetic (Linear) Sequences

General Method for Geometric $n^{th}$ Term

• Step 1: Find the common difference between terms.
• Step 2: This difference becomes the coefficient of $n$.
• Step 3: Find the "$0^{th}$" term (the term that would come before the first term) to determine the constant in the formula.

Example 1: $1, 8, 15, 22, \dots$

• Common Difference: $+7$
• $0^{th}$ Term: $1 - 7 = -6$
• Formula ($n^{th}$ term): $7n - 6$
• Finding the $50^{th}$ term:   - Substitute $n = 50$   - $7(50) - 6 = 350 - 6 = 344$
• Verification: Is 120 in the sequence?   - Set up equation: $7n - 6 = 120$   - $7n = 126$   - $n = 18$   - Yes, as $18$ is an integer.

Example 2: $4, 7, 10, 13, 16, \dots$

• Common Difference: $+3$
• $0^{th}$ Term: $4 - 3 = +1$
• Formula ($n^{th}$ term): $3n + 1$
• Verification: Is 100 in the sequence?   - $3n + 1 = 100$   - $3n = 99$   - $n = 33$   - Yes, as $33$ is an integer.
• Finding the $100^{th}$ term:   - $3(100) + 1 = 301$

Linear Sequences from Diagrams

Example: Matchstick Patterns

• Method: Write down the number of matchsticks in each image to create a numeric sequence.
• Pattern 1: 8 matchsticks
• Pattern 2: 15 matchsticks
• Pattern 3: 22 matchsticks
• Sequence: $8, 15, 22, \dots$
• Difference: $+7$
• $0^{th}$ term: $8 - 7 = 1$
• $n^{th}$ term: $7n + 1$

Geometric Sequences

Example sequence: $4, 12, 36, 108, \dots$

• Rule: Multiply by 3 each time ($\times 3$).
• Common multiple: 3.

Quadratic Sequences

Example 1: Use $n^2 - 5$ to find the first 4 terms

• Term 1 ($n=1$): $1^2 - 5 = -4$
• Term 2 ($n=2$): $2^2 - 5 = -1$
• Term 3 ($n=3$): $3^2 - 5 = 4$
• Term 4 ($n=4$): $4^2 - 5 = 11$
• Resulting Sequence: $-4, -1, 4, 11$

Example 2: Use $n^2 + 4$ to find the first 3 terms

• Term 1 ($n=1$): $1^2 + 4 = 5$
• Term 2 ($n=2$): $2^2 + 4 = 8$
• Term 3 ($n=3$): $3^2 + 4 = 13$
• Resulting Sequence: $5, 8, 13$