大象传媒

Position-to-term rules or 'nth' term

Each term in a sequence has a position. The first term is in position 1, the second term is in position 2 and so on.

Position-to-terms rules use algebra to work out the term if the position in the sequence is known. This is also called the nth term, which is a position-to-term rule that works out a term at position \(n\).

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 \(7, 10, 13, 16, 19, ...\)

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 \(4n + 2\)?