O Level MathematicsC2.7 Sequences (continuing and finding the $n^{th}$ term).

🔢 Crack the Code: Mastering Sequences & the $n^{th}$ Term!

Edudent Academy
19 Dec 25

Ever noticed how numbers can form *patterns* just like beads on a string? Sequences pop up everywhere in O-Level Maths exams, from arithmetic progressions in paper 1 to real-life problems in paper 2. **Understanding how to continue a sequence and write its nthn^{th} term formula is a high-yield skill** that saves precious time and earns easy marks.

Main Concept: Linear (Arithmetic) Sequences

A *linear* or *arithmetic* sequence increases (or decreases) by a constant difference dd. **General form:** a,  a+d,  a+2d,  a+3d, …a,\; a+d,\; a+2d,\; a+3d,\,\dots The nthn^{th} term, unu_n, is given by
un=a+(n−1)du_n = a + (n-1)d
where aa is the first term. **Bold idea:** once you know aa and dd, you can find *any* term instantly!

  • Key Point 1: Identify the common difference dd by subtracting consecutive terms.
  • Key Point 2: Substitute aa and dd into un=a+(n−1)du_n = a + (n-1)d to create the formula.
  • Problem: The sequence 3, 7, 11, 15, ... continues in the same way. Find a formula for the nthn^{th} term.
  • Step 1: Common difference d=7−3=4d = 7-3 = 4. First term a=3a = 3.
  • Step 2: Substitute into the formula
    un=3+(n−1)×4=3+4n−4=4n−1.u_n = 3 + (n-1)\times 4 = 3 + 4n - 4 = 4n - 1.
    So the nthn^{th} term is 4n−1\boxed{4n - 1}.

Practise by picking any 4-term pattern and writing its formula. **The more examples you try, the faster the pattern "clicks" on exam day.** Keep cracking those codes!