Chapter 5.4: Simple linear sequences

🎯 Learning Journey

Look for Pattern in Differences

START: Examine the sequence and find the difference between consecutive terms. In linear sequences, this difference should be constant.

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Identify Common Difference

IDENTIFY: Determine the common difference (the amount added or subtracted each time). This is the key to understanding the pattern.

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Write Rule Using Position Number

WRITE: Express the rule algebraically using the position number. For linear sequences, this follows the pattern: nth term = first term + (n-1) × common difference.

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Extend Sequence Using Rule

EXTEND: Use the rule to find additional terms in the sequence by applying the common difference or using the position number formula.

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Verify Rule Works for All Given Terms

VERIFY: Check that your rule correctly generates all the terms given in the original sequence. This confirms your rule is correct.

📖 Understanding the Topic

🎯 What You'll Learn

Linear sequences are patterns of numbers where the difference between consecutive terms is always the same. This constant difference makes the sequence predictable - once you know the pattern, you can find any term in the sequence. Linear sequences appear everywhere in mathematics and real life, from simple counting patterns to complex mathematical relationships.

🚀 Why This Matters

Predicting Costs Over Time

Many costs follow linear patterns - like subscription fees, utility bills, or savings plans where amounts increase by fixed amounts.

Planning Seating Arrangements

Venues often arrange seating in linear patterns where each row has a fixed number more (or fewer) seats than the previous row.

Scientific Measurements

Many scientific relationships follow linear patterns, allowing scientists to predict future measurements from current data.

💡 Worked Examples

Cinema charges £8 first ticket, then £6 each additional

Write first 5 terms of total cost sequence

Solution:
1st ticket: £8
2nd ticket: £8 + £6 = £14
3rd ticket: £14 + £6 = £20
4th ticket: £20 + £6 = £26
5th ticket: £26 + £6 = £32
Sequence: 8, 14, 20, 26, 32

Sequence starts 2, 8, 14, 20

What is 10th term?

Solution:
Common difference = 8 - 2 = 6
Pattern: start at 2, add 6 each time
Rule: nth term = 2 + (n-1) × 6
10th term = 2 + (10-1) × 6
= 2 + 9 × 6 = 2 + 54
Answer: 56

Pattern: 50, 45, 40, 35

When will it reach zero?

Solution:
Common difference = -5 (decreasing)
Rule: nth term = 50 + (n-1) × (-5)
For zero: 0 = 50 - 5(n-1)
5(n-1) = 50
n-1 = 10, so n = 11
Answer: 11th term

✏️ Practice Questions

Question 1: Continue sequence: 3, 7, 11, 15, ...

17, 21

19, 23

18, 22

20, 24

Question 2: Find next two terms: 20, 17, 14, 11, ...

8, 5

9, 6

7, 4

8, 6

Question 3: What is rule for sequence 5, 8, 11, 14?

Add 2 each time

Add 3 each time

Add 4 each time

Add 5 each time

⚠️ Common Mistakes & How to Avoid Them

Learn from typical errors students make and discover how to avoid them!

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Common Misconceptions

What students often do wrong:

1. Thinking rule must involve addition only: Not recognizing that sequences can decrease (subtract) or have other patterns

2. Not recognizing the pattern continues consistently: Assuming the pattern might change or not applying it correctly to find further terms

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How to Avoid These Mistakes

Correct approach: Always check the difference between consecutive terms. The pattern can be addition, subtraction, or other operations - focus on what's consistent.

Memory tip: "Same difference = linear sequence" - if the gap between terms is always the same, you've found the pattern

💡 Teacher's Tip

Write the differences between terms above the sequence. This visual approach makes the common difference obvious and helps avoid errors.

📋 Chapter Summary

🎉 Congratulations!

You've mastered Simple linear sequences!

🎯 Skills You've Developed:

✓ Recognizing and continuing linear sequences

✓ Finding the rule for linear sequences

✓ Generating sequences from given rules

✓ Using common differences to predict terms

🚀 What's Next?

Next: Learn to find missing terms in sequences using pattern recognition and systematic methods