A sequence is an ordered list of numbers. Each number is a term, and each term has a position number counting from 1. We write the nth term as Tₙ, so T₁ is the first term. There are two kinds of rule: a term-to-term rule tells you how to get from one term to the next, and a position-to-term rule gives you any term directly from its position. Exams almost always want the second kind, because it lets you jump straight to the 100th term without listing the first 99.
Students who struggle with the nth term usually do not have a problem with algebra. They have a problem with the words. If "term", "position" and "nth term" are still slightly blurry, then a question asking for T₂₀ reads as noise, and the algebra never gets a chance to help.
Ten minutes on the vocabulary makes the rest of the chapter straightforward, so it is worth doing properly before touching formulas.
Term
one number in the list
Position n
which place it sits in, from 1
Notation
Tₙ means the nth term
Term-to-term
how to get the next from the last
Position-to-term
get any term straight from n
First difference
the gap between consecutive terms
Always write the position numbers above the sequence before you do anything else. It costs five seconds and prevents the off-by-one error that ruins otherwise-correct nth term answers.
For the sequence 5, 8, 11, 14, 17, … state the first term, the fourth term, and the term-to-term rule.
Solution
n = 1, 2, 3, 4, 5 matching 5, 8, 11, 14, 17.T₁ = 5.T₄ = 14. Count positions, not gaps.8 - 5 = 3, 11 - 8 = 3, 14 - 11 = 3. The difference is constant.The step students get wrong
Confusing the position with the value. In this sequence T₃ = 11, but students frequently answer 3, reading the question as asking for the position instead of the term at that position. When you see T₃, read it aloud as "the third term" and the ambiguity disappears.
For 5, 8, 11, 14, 17, … find the position-to-term rule, then use it to find the 50th term.
Solution
3n.3n with the sequence: at n = 1, 3n = 3 but the term is 5.+2, and it is the same at every position: at n = 2, 3n = 6 and the term is 8.Tₙ = 3n + 2.n = 50: T₅₀ = 3(50) + 2 = 152.Why the position-to-term rule is worth the effort: the term-to-term rule would have needed 49 additions to reach the 50th term, with 49 chances to slip. The position-to-term rule got there in one line. This is exactly why exam questions ask for large terms: they are testing whether you found the general rule.
A term-to-term rule is not wrong, and for some sequences it is the only sensible description. The Fibonacci sequence, where each term is the sum of the previous two, has no simple position-to-term rule at O-Level. But whenever a constant difference exists, the position-to-term rule is available and is what the marks are for.
If the first differences are not constant, take the differences again. Constant second differences mean the sequence is quadratic and the rule contains an n² term. If the terms are multiplied rather than added each time, the sequence is geometric. Recognising which family you are in is the first decision in every question, and the full method is in finding the nth term of a sequence.
What does Tₙ actually mean?
It is shorthand for "the term in position n". So T₅ is the fifth term and Tₙ is the general term, the formula that works for any position. Some textbooks write aₙ or uₙ instead; they all mean the same thing.
Does a sequence have to start at position 1?
At O-Level, yes. Always number the first term as n = 1. Starting from zero shifts every formula by one and will make a correct method produce a wrong answer.
What is the difference between a sequence and a series?
A sequence is the list of terms. A series is what you get when you add the terms together. E-Maths works almost entirely with sequences; series appear in A-Maths.
Can a sequence decrease?
Yes. A constant negative difference is still a linear sequence, and the nth term simply has a negative coefficient, such as Tₙ = 20 - 3n. Everything in the method works identically.
How does this connect to the rest of E-Maths?
The position-to-term rule is a function: put a position in, get a term out. That is the same relationship you plot in the Cartesian plane, and a linear sequence plotted against its position gives a straight line whose gradient is the common difference. Recognising the standard families is covered in special sequences and pattern recognition.
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