Door 6 · Grade 8

🧩 Patterns & Sequences

An arithmetic sequence has a constant difference between consecutive terms

Number Patterns

What are Number Patterns?

A number pattern is a sequence of numbers that follows a specific rule or relationship. Each number in the pattern is called a term. Patterns help us predict future terms and understand mathematical relationships.

Real-life Example: Saving money each week - if you save ₹100 in week 1, ₹200 in week 2, ₹300 in week 3, the pattern shows you save ₹100 more each week.

What are Arithmetic Sequences?

An arithmetic sequence (or arithmetic progression) is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference (\(d\)).

Formula for Arithmetic Sequence:

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

Where:

• \(a_n\) = the \(n\)th term
• \(a_1\) = first term
• \(d\) = common difference
• \(n\) = position of the term

Examples of Arithmetic Sequences:

SequenceFirst Term (\(a_1\))Common Difference…

Worked example

Problem: Example 1: Identify if the sequence 7, 12, 17, 22, 27 is an arithmetic sequence. If yes, find the common difference and the next two terms.

- Step 1: Find difference between consecutive terms
- Step 2: \(12 - 7 = 5\)
- Step 3: \(17 - 12 = 5\)
- Step 4: \(22 - 17 = 5\)
- Step 5: \(27 - 22 = 5\)

Answer: - Step 5: \(27 - 22 = 5\)

Arithmetic Progressions (AP)

What are Geometric Patterns?

A geometric pattern (or geometric sequence) is a sequence where each term is found by multiplying the previous term by a constant. This constant is called the common ratio (\(r\)).

Formula for Geometric Sequence:

\(a_n = a_1 × r^(n-1)\)

Where:

• \(a_n\) = the \(n\)th term
• \(a_1\) = first term
• \(r\) = common ratio
• \(n\) = position of the term

Examples of Geometric Sequences:

SequenceFirst Term (\(a_1\))Common Ratio (\(r\))Next Term2, 6, 18, 54, ...2×3162100, 50, 25, 12.5, ...100×0.5 or ÷26.254, -12, 36, -108, ...4×(-3)3245, 5, 5, 5, ...5×15
What are Recursive Patterns?

A recursive pattern defines each term based on the previous term(s) using a fixed rule. Instead of giving a direct formula (like \(a_n = 2n+3\)), recursive patterns tell you how to get from one term to the next.

Recursive Formula Format:

• \(a_1 =\) [first…

Worked example

Problem: Example 1: Determine if the sequence 5, 15, 45, 135 is geometric. If yes, find the common ratio and the 8th term.

- Step 1: Check ratios between consecutive terms
- Step 2: \(15 ÷ 5 = 3\)
- Step 3: \(45 ÷ 15 = 3\)
- Step 4: \(135 ÷ 45 = 3\)
- Step 5: All ratios equal 3, so it is geometric with \(r = 3\)

Answer: - Step 5: All ratios equal 3, so it is geometric with \(r = 3\)

Geometric Progressions (GP)

What are Visual Patterns?

Visual patterns are sequences of shapes, figures, or designs that follow a predictable rule. By analyzing how the pattern changes from one figure to the next, we can predict future figures and find mathematical relationships.

Why Study Visual Patterns?

• Helps develop spatial reasoning and logical thinking
• Connects geometry with algebra (finding formulas for patterns)
• Appears in art, architecture, nature, and design
• Builds foundation for understanding functions and sequences

Common Types of Visual Patterns:

TypeDescriptionExampleGrowing PatternsFigures increase in size or number of elementsTriangle numbers, square numbersRepeating PatternsSame block repeats (tessellations, borders)ABAB pattern, striped designsRotational PatternsFigures rotate by a fixed angle each stepPinwheel designsSymmetrical PatternsFigures have reflection…

Worked example

Problem: Example 1: Draw the next figure in the pattern: Figure 1: ●, Figure 2: ●●, Figure 3: ●●● (arranged in a row). How many dots will be in Figure 10?

- Step 1: Observe the pattern: Each figure adds 1 dot to the row
- Step 2: Figure 1 has 1 dot, Figure 2 has 2 dots, Figure 3 has 3 dots
- Step 3: Figure 4 would have 4 dots: ●●●●
- Step 4: The pattern follows \(a_n = n\)
- Step 5: For Figure 10, number of dots = 10

Answer: - Step 5: For Figure 10, number of dots = 10