medium difficulty5 min

Explicit (Direct) Rule for a Sequence

An explicit rule gives the value of any term directly using its position number n.

In one line: Direct rule means jump straight to the term.

Understand it

If a sequence has a rule like an = 2n + 1, we can find the 10th term by putting n = 10 without listing all previous terms. This is useful when the question asks for a far-away term such as the 50th term.

Example

For an = 3n + 2, the 5th term is a5 = 3(5) + 2 = 17.

Formula and rule

Explicit form: an = expression in n.

Here n is a natural number showing the term position.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1Find the 12th term of an = 4n - 3.
2Put n = 12, so a12 = 4(12) - 3 = 48 - 3 = 45.

Method you can reuse

1Identify the given rule.
2Note the required position.
3Substitute n carefully.
4Simplify using correct order of operations.
5Write the term with its position.

Visual tip

A two-column table with n and an can show how position values generate sequence terms.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

For an = 2n + 1, what is the 6th term?

Think first, then show answer

The 6th term is 13 because substituting n = 6 gives a6 = 2(6) + 1 = 13.

Common wrong answer

For an = 4n - 3, the 12th term is sometimes written as 4 + 12 - 3 = 13, which.

Why it loses marks: Multiplication by n was ignored.

Don't mix this up

Students sometimes substitute the term value in place of n, but n represents the position, not the term itself.

Exam tip

Before substituting, write what n means: n = required position number.