# Number Patterns and Sequences

• Understand and apply the principle of number patterns and sequences, both linear and non-linear. 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 + (n-1)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, non-zero number called the “common ratio”. If the first term is ‘a’ and the common ratio is ‘r’, nth term = ar^n-1.

• Understand how to use the sum formula for both arithmetic progression (S = n/2(2a + (n-1)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 long-term 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 double-check your work for accuracy.