In the realm of algebra, the ability to identify and extend given number sequences stands as a cornerstone skill. This section of study notes is crafted to guide Cambridge International Examination International General Certificate of Secondary Education (CIE IGCSE) students through the intricacies of continuing sequences. Emphasizing the utility of equations for deducing subsequent terms, these notes offer a blend of theoretical insights and practical examples, tailored to foster a robust understanding of sequence continuation.

**Identifying Patterns in Sequences**

Number sequences are based on mathematical patterns. Understanding these patterns lets you find missing terms, derive formulas, and solve related problems. In this section, we'll focus on continuing sequences by identifying the numerical relationships between terms.

**Simple Differences**

Many sequences involve a constant difference between consecutive terms.

**Example:**2, 5, 8, 11, 14...

**Solution**

- Step 1: Find the difference:
- 5 - 2 = 3
- 8 - 5 = 3, etc.

- Step 2: Continue the sequence:
- 14 + 3 = 17
- 17 + 3 = 20, and so on.

**Other Arithmetic Patterns**

**Subtraction:**- Example: 80, 75, 70, 65...
- Solution: Each term is 5 less than the previous term.

**Multiplication:**- Example: 3, 9, 27, 81...
- Solution: Each term is 3 times the previous term.

**Division:**- Example: 100, 50, 25, 12.5...

**Solution:** Each term is half the value of the term before it.

**Squares, Cubes, and Beyond**

**Squares:**1, 4, 9, 16... (1², 2², 3², 4²...)**Cubes:**1, 8, 27, 64... (1³, 2³, 3³, 4³...)**Triangular Numbers:**1, 3, 6, 10... (1 + 2, 1 + 2 + 3, etc.)

**Example:** 1, 3, 6, 10, 15…

**Solution:**

- Differences between terms: 2, 3, 4, 5... (increasing by 1)
- Triangular number pattern.
- Next terms: 15 + 6 = 21, 21 + 7 = 28, etc.

**Combining Patterns**

Sequences sometimes involve multiple operations.

**Example:** 1, 4, 9, 16, 25... (Square numbers)

- Then subtract 1 from each: 0, 3, 8, 15, 24...

**Practice Questions**

**Question 1:**

Find the missing numbers: 2, 6, ___, 18, 22…

**Solution:**

- 6 - 2 = 4
- 18 - 6 = 12 (3 x 4)
- 22 - 18 = 4
- Pattern: Multiply by 3, then add 4
- Missing term: 6 x 3 = 18, 18 + 4 = 22

**Tips for Tricky Sequences**

**Alternating signs:**Consider multiplication by -1 in each step.**Fractions:**Look for division patterns.**Think outside the box:**Consider squares, cubes, prime numbers, etc.

**Question 2:**

What are the next two terms in this sequence? 5, 8, 13, 20, 29…

**Solution:**

- Differences: 3, 5, 7, 9... (odd numbers)
- Pattern: Add consecutive odd numbers.
- 29 + 11 = 40
- 40 + 13 = 53