11-Plus Mathematics: Number Sequences

Understanding Number Sequences

Number sequences are an essential part of the 11-plus Mathematics curriculum, focusing on recognizing and extending patterns. This topic helps students identify the rules governing various sequences, which is crucial for developing problem-solving skills.

Recognizing Patterns

Students learn to recognize patterns in sequences of numbers. This involves looking for common differences or ratios that can help predict subsequent numbers in the sequence.

Arithmetic Sequences

An arithmetic sequence is one where each term is generated by adding or subtracting a constant value. For example, in the sequence 2, 5, 8, 11, the common difference is 3.

Worked Example: Arithmetic Sequence

Problem: Find the next two terms in the sequence 4, 7, 10, 13.

Solution:

• The common difference is 3 (7 - 4 = 3).
• Next term: 13 + 3 = 16.
• Following term: 16 + 3 = 19.

Geometric Sequences

A geometric sequence is formed by multiplying or dividing by a constant. For instance, in the sequence 3, 6, 12, 24, each term is multiplied by 2.

Worked Example: Geometric Sequence

Problem: Determine the next two terms in the sequence 5, 10, 20, 40.

Solution:

• The common ratio is 2 (10 ÷ 5 = 2).
• Next term: 40 × 2 = 80.
• Following term: 80 × 2 = 160.

Finding Specific Terms

Students also learn how to find specific terms in a sequence using algebraic expressions. For example, the n-th term of an arithmetic sequence can be expressed as:

Tn = a + (n - 1)d

where a is the first term and d is the common difference.

Describing Patterns Algebraically

Finally, students are encouraged to describe patterns algebraically, which enhances their understanding of sequences and prepares them for more advanced mathematical concepts.

Mastering number sequences is not only vital for the 11-plus exam but also lays a strong foundation for future mathematical learning.