Sequences: deduce expressions to calculate the nth term of linear sequences

The idea

A linear sequence changes by the same amount each time. To deduce an expression for the nth term, first find the common difference between consecutive terms; this becomes the coefficient of n. For example, in 5, 8, 11, 14, ... the common difference is 3, so the expression starts as 3n. Compare the first term with 3 × 1 = 3; the sequence starts at 5, so add 2. The nth term is 3n + 2. Check by substituting n = 2: 3 × 2 + 2 = 8, which matches the second term. If the sequence decreases, the common difference is negative, so the coefficient of n is negative. Always test your expression with at least two term numbers to confirm it generates the given sequence.

What earns the marks

• Find the common difference between consecutive terms and use it as the coefficient of n.
• Compare the first term with the value given by the coefficient of n when n = 1, then adjust by adding or subtracting a constant.
• Form the expression in the form an + b, where a is the common difference and b is the constant adjustment.
• Verify the expression by substituting n = 1, 2 and 3 and checking the results match the given sequence.

Common mistakes and how to put them right

• Writing the common difference as the constant term instead of the coefficient of n.

  Put it right: The common difference belongs with n, so for 4, 7, 10, ... the expression is 3n + 1, not n + 3.

• Forgetting to include the constant adjustment after finding the coefficient.

  Put it right: For 2, 5, 8, ... the expression is 3n - 1, not just 3n.

• Assuming the first term always equals the constant.

  Put it right: The constant is found by comparing the first term with the value of an when n = 1, so it is not always the first term.

• Ignoring negative common differences.

  Put it right: For 10, 7, 4, ... the common difference is -3, so the expression is -3n + 13, not 3n + something.

Exam tips

• Always write down the common difference first; it immediately gives the coefficient of n.
• Substitute n = 1 into your expression and check it equals the first term before moving on.
• If the sequence decreases, double-check that you have used a negative coefficient for n.
• For multiple-choice questions, test each option with n = 1 and n = 2 rather than trying to spot the answer.