How to Work Out the Next Number in a Sequence

Sequence questions look harder than they are. The hard part is spotting the rule; the rest is arithmetic. Once you know whether you are adding or multiplying, everything follows from two formulas.

Find the rule before you find the term

Write the sequence down and put the gaps underneath it. Take 3, 7, 11, 15, 19. The gaps are 4, 4, 4, 4. Constant difference, so it is arithmetic with first term a = 3 and common difference d = 4. The nth term is 3 + 4(n - 1), which tidies up to 4n - 1. The 20th term is 4 x 20 - 1 = 79, and you never had to write out the eighteen terms in between.

Now take 5, 15, 45, 135. The differences are 10, 30, 90, which are not constant, so try dividing instead. 15 divided by 5 is 3, and 45 divided by 15 is 3. Geometric, ratio 3. The 10th term is 5 x 3⁹ = 5 x 19,683 = 98,415. That is the thing about geometric sequences. They get large in a hurry.

When the difference is not constant

If neither the differences nor the ratios settle down, take the differences of the differences. For 2, 5, 10, 17, 26 the first differences are 3, 5, 7, 9 and the second differences are all 2. A constant second difference means the rule is quadratic, and in this case it is n² + 1. Two other patterns are worth recognising on sight: Fibonacci, where each term is the sum of the two before it (1, 1, 2, 3, 5, 8, 13, 21, 34, 55), and factorials, where 1, 2, 6, 24, 120 is each term multiplied by the next counting number.

Using the calculator, and what to type where

Enter the terms you already have, in order, separated by commas. Do not skip a term in the middle and expect the calculator to guess: a gap changes the differences and the rule that comes back will be wrong. Then tell it which term you want. Ask for the 50th term and you get a single number. Ask for the sum and you get the total of every term up to that point.

Sum versus nth term

These are different questions and people mix them up constantly. For 3, 7, 11, 15 the 20th term is 79. The sum of the first 20 terms is 20 divided by 2, times (3 + 79), which is 10 x 82 = 820. If a question asks how much you have saved in total after 20 months, you want the sum. If it asks what you saved in month 20, you want the term.

Where sequences show up outside a maths lesson

Geometric sequences are compound interest wearing a different hat. A balance growing 5% a year is a geometric sequence with r = 1.05, which is exactly why compound interest feels slow for a decade and then suddenly does not. Arithmetic sequences turn up in loan amortisation, brick courses and anything where you add a fixed amount each step. Halving patterns, like a drug clearing your system, are geometric with r = 0.5.

If you are working through a homework set, the sequence tool sits alongside the rest of the free maths calculators, and checking your rule there is faster than recomputing by hand.

Common questions

How many terms do I need before the calculator can find the pattern? Four is a sensible minimum and five is better. With three terms there is often more than one rule that fits, and the calculator picks the simplest, which may not be what your textbook wants.

What is the difference between a sequence and a series? A sequence is the list of terms. A series is what you get when you add them up. So 3, 7, 11 is a sequence and 3 + 7 + 11 = 21 is the series.

Can a sequence have a negative common difference? Yes. In 20, 17, 14, 11 the common difference is -3, and the formula works exactly the same way. The 12th term is 20 + (-3)(11) = -13.

Why does my sequence not match any standard type? Some sequences are defined by position rather than by a step rule, such as square numbers or primes. Check whether the terms are recognisable in their own right before hunting for a difference that is not there.