Year 9 Maths The Thai Guy

Lesson 12.1 · 25 min

Sequences

Term-to-term and position-to-term rules, and finding the nth term of linear and quadratic sequences.

In this lesson you will learn to…

  • Generate terms from a term-to-term rule and from a position-to-term rule
  • Find the nth term of an arithmetic sequence
  • Use the nth term to find any term, however far along
  • Find the nth term of a simple quadratic sequence

Two kinds of rule

There are two ways to describe a sequence, and exam questions use both words, so it is worth being clear about the difference.

A term-to-term rule says how to get from one term to the next: add 5, double it, square it then subtract 1. To use it you must know where to start, and to find the 50th term you would have to work through the 49 before it.

A position-to-term rule — the nth term — takes the position number and gives you the term directly. Want the 50th term? Put n=50n = 50 in and you are done.

Finding the nth term of an arithmetic sequence

An arithmetic sequence goes up (or down) by the same amount each time. That amount is the common difference, dd.

The method most students find quickest:

  1. Find the common difference dd. Subtract any term from the one after it.
  2. Write down dndn. This is the “skeleton” of the answer.
  3. Compare dndn with the real sequence at the same positions.
  4. Add or subtract whatever constant makes them match.
  5. Check with a term you have not used yet.

The formal version of the same thing is

nth term=a+(n−1)d\text{nth term} = a + (n-1)d

Both give the same answer. The formula is safer when the numbers are awkward; the comparison method is faster when they are not.

Watch the signs

If the sequence goes down, dd is negative, and the nth term will look like 20−2n20 - 2n rather than 2n+202n + 20. The commonest mistake in the whole topic is dropping that minus sign when expanding (n−1)d(n-1)d.

Quadratic sequences

Not every sequence has a constant difference. Take −6,−3,2,9,18,…-6, -3, 2, 9, 18, \dots

First differences: 3,5,7,93, 5, 7, 9 — not constant. Second differences: 2,2,22, 2, 2 — constant.

When the second differences are constant, the sequence is quadratic: its nth term involves n2n^2.

For the sequences you meet at this level, the second difference is 2, which means the n2n^2 coefficient is 1. So you can find the rule by simply writing the square numbers above the sequence and spotting the constant gap between them:

nn12345
n2n^21491625
sequence−6-6−3-32918
difference−7-7−7-7−7-7−7-7−7-7

The gap is −7-7 every time, so the nth term is n2−7n^2 - 7.

Always check. Substitute a position you did not use while working it out. If n=6n = 6 gives 36−7=2936 - 7 = 29, and the sequence continues 18,2918, 29, you are right.

Key rules and formulas

  • Term-to-term rule

    Tells you how to get from one term to the next. You need a starting term to use it. Example: square then subtract 1, starting at 2, gives

  • Position-to-term rule (the nth term)

    Tells you how to get a term straight from its position, without working through the ones before it. Much more useful for questions like "find the 100th term".

  • The nth term of an arithmetic sequence

    where is the first term and is the common difference. Expand the brackets and simplify to get the tidy form.

  • The shortcut most people use

    The nth term of an arithmetic sequence is always , where is the common difference. Find , write , then work out what to add or subtract to make the first term come out right.

  • Quadratic sequences

    If the differences are not constant, look at the second differences. If those are constant, the sequence is quadratic. Compare it term by term with the square numbers

Worked examples

Worked example 1

Finding the nth term of an arithmetic sequence

Find the nth term of

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Using the formula instead

Find the nth term of using .

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

A quadratic sequence

Find the nth term of

Try it yourself first, then reveal the solution one step at a time.

Practice: sequences

10 questions

Ten questions. The algebra boxes accept any equivalent form — is just as right as .
  1. Question 1 of 10Type the answerFoundation
    The term-to-term rule is add 6 and the first term is 5. What is the fourth term?
  2. Question 2 of 10Type the answer
    The term-to-term rule is square then subtract 1, and the first term is 2. What is the third term?
  3. Question 3 of 10Complete the table
    The position-to-term rule is . Complete the table.
    Position, 123410
    Term
  4. Question 4 of 10Algebraic answer
    Find the nth term of

    Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.

  5. Question 5 of 10Algebraic answer
    Find the nth term of

    Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.

  6. Question 6 of 10Type the answer
    A sequence has nth term . Find the 50th term.
  7. Question 7 of 10Put in order
    Put the steps for finding the nth term of an arithmetic sequence into the right order.

    Put these in the right order. Drag them, or use the arrow buttons.

    1. Write down
    2. Add or subtract the constant that makes them match
    3. Find the common difference,
    4. Compare with the sequence at the same positions
    5. Check using a term you have not already used

  8. Question 8 of 10Multiple choice
    The first differences of a sequence are . What kind of sequence is it?
  9. Question 9 of 10Algebraic answerChallenge
    Find the nth term of the quadratic sequence

    Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.

  10. Question 10 of 10Choose all that applyChallenge
    A sequence has nth term . Select every number that appears in this sequence.

    Select every correct answer — there may be more than one.