Number Patterns and Sequences

Understand and apply the principle of number patterns and sequences, both linear and nonlinear. This involves discerning a particular rule that dictates the relation between consecutive numbers or terms in the sequence.

Linear sequences progress by a steady rate called the “common difference”. This can be found by subtracting any term from the term succeeding it.

The nth term of a linear sequence can be identified by the formula: a + (n1)d, where ‘a’ is the first term, ‘d’ is the common difference, and ‘n’ is the placement of the term in the sequence.

Quadratic sequences feature a constant second difference, meaning the difference between consecutive terms changes at a steady rate.

To find a formula for the nth term in a quadratic sequence, create a new sequence from the differences between successive terms, repeat until your sequence shows a constant difference, note the difference and create an nth term expression e.g. 2n^2  3n + 5.

Arithmetic progression is a linear sequence where every term after the first is obtained by adding a constant difference to the preceding term. If the first term is ‘a’ and the common difference is ‘d’, nth term = a + (n  1)d.

Understand the concept of geometric sequences. In geometric progressions, each term after the first is found by multiplying the previous term by a fixed, nonzero number called the “common ratio”. If the first term is ‘a’ and the common ratio is ‘r’, nth term = ar^n1.

Understand how to use the sum formula for both arithmetic progression (S = n/2(2a + (n1)d)) and geometric progression (S = a(rn − 1) / r  1).

Familiarize yourself with Fibonacci sequences where each term is the sum of the two preceding ones, often starting with 0 and 1.

Practice your skills of analysis, application, and synthesis by creating your own number patterns and sequences, predicting future terms and understanding how alterations to the rule can change the sequence.

Remember to review common pitfalls, misconceptions, and common test mistakes when working with patterns and sequences. This includes confusing terms in a sequence with their position, forgetting that sequences can decrease, and neglecting to check if a sequence is arithmetic or geometric.

Utilise past papers and tailored exercises to commit sequence principles to longterm memory and improve fluency when manipulating sequences.

Remember to always check your work. A simple calculation mistake can disrupt the entire pattern or sequence, so doublecheck your work for accuracy.