Unit 1
Patterns and Sequences
About this unit
Identify rules in number sequences, repeating patterns, and arithmetic progressions. Learn the cycle-remainder shortcut for repeating sequences and the sum formulas for arithmetic series.
What types of questions will you face?
- 1Find the element at a large position in a repeating sequence using the cycle-remainder shortcut (divide position by cycle length; the remainder locates the answer in the cycle)
- 2Calculate the nth term of an arithmetic sequence given the first term and common difference (nth term = first + (n−1) × difference)
- 3Find the sum of an arithmetic series — such as all odd numbers from 1 to 99, or all multiples of 3 up to a limit — using the sum shortcut
- 4Identify the rule in a non-linear or second-difference sequence and extend it step by step to find a later term
- 5Work out the total number of items in a repeating multi-colour or multi-type pattern when you are told how many of one element there are (e.g. bead bracelets, knitting rows, page layouts)
Skills you will build
- Applying the cycle-remainder method to pinpoint any term in a repeating pattern without listing every element
- Writing and using the nth-term formula (a + (n−1)d) for arithmetic sequences quickly under time pressure
- Calculating arithmetic series sums using the formula n ÷ 2 × (first + last) to avoid slow term-by-term addition
- Spotting second-order differences (the gaps between gaps increase by a fixed amount) and extending sequences accurately
- Translating a word-based pattern description into a structured cycle length and solving total-count problems from it
By the end of this unit, you will be able to
- Find any term of a repeating sequence in seconds using division and remainder, without listing every element
- Calculate any specific term or partial sum of an arithmetic sequence accurately and quickly
- Identify whether a sequence is linear (arithmetic), exponential, or requires second-difference analysis — and apply the right method
- Solve total-count problems for multi-element repeating patterns by working backwards from the given count of one element
Difficulty profile
Questions in this unit range from Very Easy to Medium. Repeating patterns and simple arithmetic sequences are Very Easy to Easy and reward knowing the shortcut. Second-difference sequences and triangle-pattern questions are Medium and appear in the mid-section of the OC MR paper.
Exam tip: Patterns and Sequences
For any repeating sequence, divide the position number by the cycle length and use only the remainder. If the remainder is 0, the answer is the last element of the cycle — not the first. For series sums, count the terms first using (last − first) ÷ step + 1, then apply n ÷ 2 × (first + last). Practise both reflexes until they are instant.
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