Generating arithmetic (linear) sequences
Arithmetic sequences are everywhere — from the seating rows in a stadium to the pattern of numbers on a bus timetable. They’re just numbers that grow (or shrink) by the same amount each time, and learning how to generate them helps us spot and describe these everyday patterns. Jump to the questions
Practise now
Enter the first 5 terms for each sequence.
Topic guide
What this worksheet practises
This worksheet provides practice on generating the terms of an arithmetic (or linear) sequence when you are given its "nth term" formula. This is the opposite of finding the formula from a given list of numbers.
Key method
The "nth term" formula (like 3n + 2) is a rule that tells you exactly what number sits in position 'n' of the sequence.
- To find the 1st term, substitute n = 1 into the formula.
- To find the 2nd term, substitute n = 2 into the formula.
- To find the 3rd term, substitute n = 3 into the formula.
- To find the 50th term, substitute n = 50 into the formula.
- Always remember that a number next to a letter means multiply (so 3n means 3 × n).
Worked example
The nth term of a sequence is 4n − 3. Find the first three terms, and the 10th term.
Step 1: Find the 1st term (n = 1).
4(1) − 3 = 4 − 3 = 1.
Step 2: Find the 2nd term (n = 2).
4(2) − 3 = 8 − 3 = 5.
Step 3: Find the 3rd term (n = 3).
4(3) − 3 = 12 − 3 = 9.
The first three terms are 1, 5, 9.
Step 4: Find the 10th term (n = 10).
4(10) − 3 = 40 − 3 = 37.
Common mistakes to avoid
The most common mistake is confusing 'n' with the actual sequence numbers. If asked if 25 is in the sequence 4n - 3, you do not substitute n = 25. You set up an equation (4n - 3 = 25) and solve for 'n' to see if 'n' is a whole integer.
How to check your answer
Because the formula is linear (there are no squared powers), your generated sequence should go up or down by the exact same amount every time. In our example (1, 5, 9...), the numbers go up by 4 each time. This perfectly matches the "4n" part of our formula. If the gaps aren't constant, you have made a calculation error.