Using the nth term

If the nth term of a sequence is known, it is possible to work out any number in that sequence.

Example

Write the first five terms of the sequence 3n + 4.

n represents the position in the sequence. The first term in the sequence is when n = 1, the second term in the sequence is when n = 2, and so on.

To find the terms, substitute n for the position number:

  • when n = 1, 3n + 4 = 3 \times 1 + 4 = 3 + 4 = 7
  • when n = 2, 3n + 4 = 3 \times 2 + 4 = 6 + 4 = 10
  • when n = 3, 3n + 4 = 3 \times 3 + 4 = 9 + 4 = 13
  • when n = 4, 3n + 4 = 3 \times 4 + 4 = 12 + 4 = 16
  • when n = 5, 3n + 4 = 3 \times 5 + 4 = 15 + 4 = 19

The first five terms of the sequence: 3n + 4 are 7, 10, 13, 16, 19, ...

Quadratic sequences

The nth term for a quadratic sequence has a term that contains x^2. Terms of a quadratic sequence can be worked out in the same way.

Example

Write the first five terms of the sequence n^2 + 3n - 5.

  • when n = 1, n^2 + 3n - 5 = 1^2 + 3 \times 1 -  5 = 1 + 3 – 5 = -1
  • when n = 2, n^2 + 3n - 5 = 2^2 + 3 \times 2 -  5 = 4 + 6 – 5 = 5
  • when n = 3, n^2 + 3n - 5 = 3^2 + 3 \times 3 -  5 = 9 + 9 – 5 = 13
  • when n = 4, n^2 + 3n - 5 = 4^2 + 3 \times 4 -  5 = 16 + 12 – 5 = 23
  • when n = 5, n^2 + 3n - 5 = 5^2 + 3 \times 5 -  5 = 25 + 15 – 5 = 35

The first five terms of the sequence: n^2 + 3n - 5 are -1, 5, 13, 23, 35

Working out terms in a sequence

When the nth term is known, it can be used to work out specific terms in a sequence. For example, the 50th term can be calculated without calculating the first 49 terms, which would take a long time.

Question

What is the 100th term in the sequence 5n - 3?

To answer this, the position is 100, so substitute n for 100.

5n - 3 = 5 \times 100 - 3 = 500 - 3 = 497

497 is the 100th term in the sequence 5n - 3.

Question

Is the number 14 in the sequence 4n + 2?

To work out whether 14 is in this sequence, put the nth term equal to the number and solve the equation.

\begin{array}{ccc} 4n + 2 & = & 14 \\ -2 && -2 \end{array}

\begin{array}{ccc} 4n & = & 12 \\ \div 4 && \div 4 \end{array}

n = 3

This means that 14 is in the sequence and it is the third term.