How do you tell what a sequence is saying?
You see sequences whenever numbers follow a rule: scores going up each round, followers increasing each week, or a pattern in tiles. What catches people out is that the position matters just as much as the value.
A sequence is a list of numbers written in a fixed order. Each number is called a term, and where it sits in the list tells you which term it is.
For example, in 1, 3, 5, 7, 9, ..., the numbers are not just floating around randomly. You can talk about the first term, second term, third term and so on. You write these as u₁, u₂, u₃, ..., and the general term is uₙ.
The small number underneath, called the index, tells you the term's position. So in u₅, the index is 5.
Using subscripts is way better than naming terms with letters like a, b, c, because u₂₀ instantly tells you that you mean the 20th term. In sequence questions, n is a position number, so you use whole numbers only.
Sometimes a question starts at u₀ instead of u₁. That changes every term number, so check the starting point before you do anything.
A useful comparison is this: a graph line is like a full song waveform with every point available, but a sequence is more like only the beats you sampled at 1, 2, 3, 4, ... . You only use specific slots, not every possible value in between.
How can you jump straight to any term?
If a question asks for the 50th term, you do not want to build all 49 terms before it. That is where a direct rule helps.
An explicit formula gives you the value of a term from its position number n in one step.
For odd numbers, the rule is
Here, n is the term number. Since term numbers count positions, you do not substitute decimals like 2.5.
Method: using an explicit rule
- Pick the term number you want.
- Substitute that value into the formula.
- Simplify carefully.
Worked example: check the rule
For
- First term:
- Second term:
- Fifth term:
So the rule matches the sequence 1, 3, 5, 7, 9, ...
A lot of sequences you meet are linear sequences, and their direct rule often looks like
That looks similar to y = ax + b, but there is a difference. A function can usually take any real input, while a sequence only uses counting-number inputs like 1, 2, 3, ... . So your sequence is discrete, not continuous.
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