This is Chapter 10 – Number Sequences for the Year 11 Mathematical Methods course.

This chapter develops a structured approach to numerical patterns and introduces formulas that describe how quantities change from one term to the next. Students learn to recognise whether a sequence follows an additive, multiplicative or more general rule and to represent that rule algebraically.

We begin with general number sequences and the language used to describe individual terms. Arithmetic sequences are then introduced as patterns with a constant difference, allowing students to determine unknown terms and construct formulas for the nth term.

Geometric sequences replace constant difference with constant ratio. Students compare additive and multiplicative change and learn to recognise which model is appropriate from numerical, algebraic or contextual information. This distinction later connects closely with linear and exponential growth. Students use both recursive descriptions and explicit formulas, building an understanding of the difference between defining how a sequence continues and calculating a distant term directly.

Series are then introduced as the sums of sequence terms. Arithmetic and geometric series require students to distinguish carefully between a term and a total, select the appropriate formula and interpret each variable correctly within a problem.

Sequences and series appear in financial modelling, repeated growth, instalments and many other processes that occur in discrete stages. They also strengthen algebraic pattern recognition and provide useful preparation for exponential functions and later calculus, where students examine change across continuous rather than discrete intervals. Clear interpretation is important when a sequence represents a real process, because the first term, common difference or common ratio must be matched correctly to the situation.

Subchapters:

• 10A: Number sequences
  
• 10B: Arithmetic sequences
• 10C: Geometric sequences
• 10D: Series
• 10E: Arithmetic series
• 10F: Geometric series

By the end of this chapter, students will be able to identify, represent and analyse arithmetic and geometric sequences and calculate corresponding series using appropriate algebraic models.