大象传媒

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 \(5n - 3\)?

Question

Is the number 14 in the sequence \(4n + 2\)?