nth Term of Linear Sequences

A sequence is a list of numbers in order. Each number is a term. The position of a term says where it is in the list: 1st, 2nd, 3rd and so on.

In the sequence 4, 9, 14, 19, \ldots the 3rd term is 14. Its position is 3 and its value is 14

In a linear sequence, also called an arithmetic sequence, you add or subtract the same number each time. This number is the common difference. In 4, 9, 14, 19, \ldots it is 5, so the next term is 19 + 5 = 24

The nth term is a rule, in terms of n, that gives any term from its position. Put n = 1 to get the 1st term, n = 2 to get the 2nd term, and so on.

In the exam you are usually asked to work out the next term and explain how you found it (1 mark), to work out a term from an nth term (1 mark), to find an expression, in terms of n, for the nth term (2 marks), or to decide whether a number is in a sequence and give a reason (1 or 2 marks).

The number in front of n is the common difference. So a sequence that goes up by 3 each time has an nth term that starts 3n

List the values of 3n: 3, 6, 9, 12, \ldots Then compare each term with the value of 3n in the same position. Each term is the same amount more than 3n, or the same amount less.

A quick way to find that number:

\text{1st term} - \text{common difference}

If the answer is negative, the nth term has a subtraction at the end.

If the terms go down, you subtract the same number each time. In 30, 26, 22, 18, \ldots you subtract 4 each time, so the next term is 18 - 4 = 14. Keep going and the terms become negative: \ldots, 6, 2, -2, -6

To explain how to find the next term, give the number as well as the direction, such as "subtract 4" or "go down by 4". Writing "it goes down" on its own is not enough.

The difference between two terms further apart is the number of steps times the amount you subtract. From the 2nd term, 26, to the 5th term, 14, is 5 - 2 = 3 steps, and 3 \times 4 = 12

Sometimes you are given an nth term that subtracts, such as 40 - 3n. Use it in the same way: the 5th term is 40 - 3 \times 5 = 25

Make the nth term equal to the number and solve for n. If n is a whole number, the number is in the sequence and n is its position. If n is not a whole number, the number is not in the sequence.

Another way is to write out the terms until you go past the number. Then show the two terms either side of it.

Always say yes or no, and show the working that proves it. Writing down the nth term on its own does not answer the question.

Some sequences are patterns of dots, sticks or tiles. The pattern number is the position n, and the number of shapes is the term.

Count the shapes in the first few patterns. The number added each time goes in front of n. Here are the first three patterns of a sequence. Worked example 6 uses them.

• Writing n + 8 for a common difference of 8. The difference multiplies n, so it starts 8n.
• Stopping at 8n. Compare 8n with the terms and add or subtract the number that makes them match.
• Giving the next term or "add 5". The question wants an expression in terms of n, such as 5n - 2.
• Swapping the numbers, such as -2n + 5 for 5n - 2. Check your rule by putting n = 1 and n = 2 back in.
• Mixing up a term's position and its value. The 6th term is found with n = 6; its value is usually not 6.
• Saying a number is or is not a term with no working. Solve the nth term equal to the number, or list the terms either side, then say yes or no.