Sequences

Introduction to Sequences

  • A sequence is a list of numbers that often follow a particular pattern.
  • These numbers are also called the terms of the sequence.

Arithmetic Sequences

  • An Arithmetic Sequence is a sequence where the difference between consecutive terms is constant. This difference is known as the common difference.
  • The formula to find the nth term of an arithmetic sequence is a + (n - 1)d, where a is the first term and d is the common difference.
  • For example, in the sequence 2, 4, 6, 8, the common difference is 2.

Geometric Sequences

  • A Geometric Sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio.
  • The formula to find the nth term in a geometric sequence is ar^(n-1), where a is the first term and r is the common ratio.
  • For example, in the sequence 3, 6, 12, 24, the common ratio is 2.

Sequences of Squares, Cubes and Triangular Numbers

  • The sequence of square numbers follows the pattern n^2, e.g. 1, 4, 9, 16 are the first four square numbers.
  • The sequence of cube numbers follows the pattern n^3, e.g. 1, 8, 27, 64 are the first four cube numbers.
  • A sequence of triangle numbers follows the pattern of adding the next natural number, e.g. 1, 3 (1+2), 6 (1+2+3), 10 (1+2+3+4) are the first four triangle numbers.

Finding the nth Term of Linear Sequences

  • In a linear sequence, the difference between each term is constant.
  • The formula for the nth term is usually in the form an + b.
  • For example, in the sequence 3, 5, 7, 9 the nth term would be 2n + 1, where n represents the position of each term in the sequence.

Fibonacci Sequence

  • Fibonacci sequence is a series of numbers where a number is the addition of the last two numbers, starting with 0 and 1. The sequence goes: 0, 1, 1, 2, 3, 5, 8, 13…
  • It is named after Italian mathematician, Leonardo Fibonacci.

Summary

  • Familiarise yourself with the patterns in different types of sequences.
  • Revision and practice are key to recognising and using these patterns to calculate the nth term of different sequences.
  • Remember the formulas to find the nth term for both arithmetic and geometric sequences.
  • Being able to identify and work with various sequences is fundamental in solving more complex algebraic problems.