Finding The Nth Term Worksheet

Working with Sequence Patterns

Most people approach these problems by listing terms and staring at them until something clicks. That works sometimes. It doesn't work often enough. Here's the actual process. Take the sequence. Write down the differences between consecutive terms. If those first differences are constant, you're dealing with an arithmetic sequence and the nth term follows the form an + b. The value of a is simply that common difference. Plug in n = 1, match it against the first term, and solve for b. Done. If the first differences aren't constant, take the differences of the differences. A constant second difference means you're looking at a quadratic sequence, and the nth term takes the form an² + bn + c. The value of a is half the second difference. From there, subtract an² from each term to leave a linear remainder, then solve for b and c the same way you would for any arithmetic sequence.

I ran into this last year with a set of worksheet problems where the sequence was defined recursively rather than explicitly. Students were given f(n) = f(n-1) + 2n and asked to find the nth term. The standard difference-table approach still works but nobody teaches it that way. You compute the first few terms—3, 8, 15, 24—then run the difference method on those. Second difference comes out constant at 2, so a = 1. Subtract n² from each term and you get 2, 4, 6, 8, which is just 2n. The nth term is n² + 2n - 1. I wish more resources covered recursive definitions before students hit the standard drills.

Download Your Finding The Nth Term Worksheet

The files circulate on mostly UK-based education sites like BBC Bitesize, Corbettmaths, and some teacher resource repositories. Search for "nth term worksheet pdf" and you'll find sets ranging from basic arithmetic sequences through quadratic and even mixed-pattern challenges. Most are free. A few require an account signup. I've used a handful across different year groups and the quality varies quite a bit. Some worksheets include square numbers, cube numbers, and triangular number sequences woven in. Those are worth practicing because they show up when the pattern isn't immediately obvious as arithmetic or quadratic. A triangular number sequence like 1, 3, 6, 10, 15 has first differences of 2, 3, 4, 5 and a constant second difference of 1, giving you nth term ½n² + ½n. Students who memorize these five or six standard sequences save themselves a lot of time during exams. One thing I've noticed that causes consistent trouble: sign errors when the common difference is negative. Take the sequence 10, 7, 4, 1, -2. The first difference is -3, so a = -3. Working through n = 1 gives -3 + b = 10, so b = 13. The nth term is -3n + 13. Students frequently write 3n + 13 or -3n - 13 and then wonder why plugging in n = 1 doesn't reproduce the original sequence. Always check your formula against at least two terms before moving on. It takes five seconds and catches the majority of mistakes.

Another pitfall involves sequences where c turns out to be zero. If you have 2, 6, 12, 20, the second difference is 2, so a = 1. Subtract n² and you get 1, 2, 3, 4, meaning b = 1 and c = 0. The nth term is n² + n. Some students insist on writing + 0c or getting confused about whether c exists. It's fine if it's zero. Just state the formula cleanly. The method breaks down for non-polynomial sequences. Geometric sequences like 2, 6, 18, 54 follow a completely different rule—nth term is a × r^(n-1)—and the difference table approach tells you nothing useful. Oscillating sequences, prime-number-based patterns, and anything defined by modular arithmetic also fall outside this framework. If a worksheet includes these, the expected approach is usually pattern recognition by inspection or identifying the operation applied at each step rather than fitting a polynomial. For most classroom settings, a student who can reliably execute the difference method on arithmetic and quadratic sequences within two minutes per problem is working at the expected pace. The whole process from reading the sequence to verifying the formula should take roughly 90 seconds if you know what you're doing. The bottleneck is usually algebraic manipulation, not the sequence logic itself.