Sequence Calculator for GCSE and A-Level Maths

A sequence is an ordered list of numbers generated by a rule. In UK maths courses, especially GCSE and A-Level, you are expected to identify the pattern quickly, write the nth term clearly, and use sum formulas accurately. This page combines a practical calculator with a complete revision guide, so you can move from pattern spotting to full marks methods in one place.

When students lose marks on sequence questions, it is rarely because the concept is impossible. More often, marks are lost through small slips: mixing up the first term with the nth term, using the right formula with the wrong value of n, forgetting brackets around n-1, or applying the geometric sum-to-infinity rule when the ratio does not converge. This guide is designed to remove those errors. You can enter your own terms, let the calculator identify the likely type, and then compare its explanation with your working line by line.

The calculator is intentionally exam-focused. It prioritises arithmetic, geometric, Fibonacci, quadratic and triangular patterns because these cover the majority of sequence questions you will meet in UK specifications. It also provides any requested term, so if your teacher asks for the 30th term, the 100th term, or a check against your manual answer, you can test it instantly and still review the algebra behind the number.

Use the tool first for speed, then read the examples and method notes below for understanding. That combination gives the strongest revision result: confidence in your result and confidence in your method.

Interactive Sequences Calculator

Enter the first few terms separated by commas. The JavaScript calculator will identify the sequence type where possible, display the nth term formula, and compute any term you request. For arithmetic and geometric sequences it also shows the sum of the first n terms, and for convergent geometric series it shows the sum to infinity.

Required formulas used: Arithmetic nth term = a+(n-1)d, arithmetic sum = n/2(2a+(n-1)d). Geometric nth term = ar^(n-1), geometric sum = a(1-r^n)/(1-r), and geometric sum to infinity = a/(1-r) when |r|<1 (convergent).

Core Formula Reference

Formula memory matters, but method discipline matters more. In an exam, write your substituted values clearly before simplifying. If you place the right values under the right symbols, the rest is routine algebra. If you skip that step, it is easy to make a sign error or lose a bracket and produce a wrong final value even when you know the right formula.

Arithmetic Sequences: Rule, Nth Term and Sums

An arithmetic sequence changes by adding the same amount every time. That fixed amount is the common difference, written as d. If the first term is a, the nth term is a+(n-1)d. This formula is fundamental for GCSE algebra and remains important at A-Level because it sets up clean linear modelling and progression work. The structure is simple: start at the first term and move forward n-1 steps, adding d each step.

Take the requested example: 3, 7, 11, 15. The common difference is d=4. Using a=3 and d=4 in a+(n-1)d gives 3+4(n-1)=3+4n-4=4n-1. So the nth term can be written as 4n-1. In many exam questions, you gain marks both for identifying d and for presenting the nth term in a simplified form. Writing both can be a strong way to secure method marks and reduce ambiguity.

Once you have the nth term, individual terms become quick checks. For the 25th term in this sequence, substitute n=25 into 4n-1. You get 100-1=99. The same answer comes from repeated addition, but the formula is faster and less error-prone. This is exactly why examiners like nth term questions: they test whether you can turn pattern recognition into algebra rather than rely on manual continuation.

The sum formula for arithmetic sequences is n/2(2a+(n-1)d). It comes from pairing terms from the start and end, but you do not need to re-derive it in every exam. You do need to apply it carefully. If a=3, d=4 and n=20, then S20=20/2(2·3+19·4)=10(6+76)=820. A common mistake is to use n instead of n-1 in the bracket; another is forgetting that 20/2 multiplies the entire bracket expression. Keep brackets visible until your final simplification.

Arithmetic models appear in budgeting, staircase design, regular payment plans and any linear growth context. In UK exam papers, contextual wording might hide the sequence language, but the algebra remains the same. If each stage increases by the same amount, you are in arithmetic territory.

Geometric Sequences: Growth, Decay and Convergence

A geometric sequence changes by multiplication rather than addition. If each term is multiplied by a fixed ratio r, and the first term is a, then the nth term is ar^(n-1). This is central for exponential growth and decay, compound percentage change, depreciation and many modelling tasks at A-Level. The pattern is multiplicative, so differences are not constant, but ratios are.

For example, in 5, 10, 20, 40, the ratio is r=2 and a=5. The nth term is 5·2^(n-1). To find the eighth term, substitute n=8: 5·2^7=640. If the ratio is fractional, the sequence may decrease while staying positive, such as 48, 24, 12, 6 where r=1/2. If the ratio is negative, signs alternate, such as 12, -6, 3, -1.5 with r=-1/2. The same nth term rule still works.

The finite sum formula is a(1-r^n)/(1-r) when r ≠ 1. If r=1, every term is a and the sum is simply na. Students often reverse numerator signs or denominator signs; remember that equivalent forms exist, but keep your version consistent to avoid sign slips. Substituting values in one clear line is usually enough for full method credit.

The sum to infinity is a special case: a/(1-r) when |r|<1. This condition is essential. If |r| is less than 1, powers of r get smaller and approach zero, so the series is convergent. If |r| is greater than or equal to 1, terms do not settle toward zero and the infinite sum is not finite. In short, always check the magnitude of r before writing an infinite sum result.

Example with convergence: a=9 and r=1/3. The sum to infinity is 9/(1-1/3)=9/(2/3)=13.5. Example without convergence: a=4 and r=1.2. Because |r|>1, the terms grow and there is no finite sum to infinity. This is a common one-mark checkpoint on UK exam papers: state convergence condition before calculation.

Geometric reasoning connects directly to logs and exponentials in higher-level work. Practising sequence questions early makes those later topics easier because the same ideas of repeated multiplication and powers are already familiar.

Fibonacci Sequence in Exam Language

The Fibonacci sequence is defined recursively: each term equals the sum of the previous two terms. A common start is 1, 1, then 2, 3, 5, 8, 13 and so on. Because the rule uses two earlier terms, Fibonacci is neither arithmetic nor geometric. Its differences are not constant and its ratios are not constant, although ratios do approach a limit as n grows.

In GCSE tasks, Fibonacci style questions usually test whether you can follow and extend a recursive rule accurately. In A-Level contexts, recursion can appear in broader modelling, proof or algorithm questions. If asked for a specific term, an efficient approach is iterative calculation rather than writing out every intermediate number. The calculator on this page follows the same practical method, especially for larger n values where manual continuation is slow and error-prone.

A useful habit is checking parity and size as you calculate: if the sequence suddenly shrinks or repeats unexpectedly without reason, you probably made an arithmetic slip in an earlier step. Recurrence questions reward careful arithmetic discipline more than advanced algebra, so neat line-by-line working often secures marks quickly.

Quadratic Sequences and Constant Second Difference

Quadratic sequences are usually modelled by an^2+bn+c. The hallmark is that first differences are not constant, but second differences are constant. This gives a fast diagnostic test in exams: write the terms, calculate first differences, then calculate differences of those differences. If the second row is constant, you have a quadratic sequence.

Example: 1, 4, 9, 16, 25. First differences are 3, 5, 7, 9. Second differences are 2, 2, 2, so the sequence is quadratic. The nth term here is n^2. Another example: 2, 6, 12, 20, 30. First differences 4, 6, 8, 10 and second differences 2, 2, 2 again show a quadratic rule. The calculator estimates coefficients and returns a formula form to help you check your own derivation.

For many GCSE higher-tier questions, it is enough to identify the nth term by comparing with known square-based patterns. At A-Level, you may be asked to derive coefficients systematically. The second difference gives 2a, so a is half the constant second difference. Then use early terms to solve for b and c. This method is reliable and exam-safe when presented clearly.

The key line to remember is exactly what examiners expect: quadratic sequences have constant second differences. Writing this sentence in your solution often earns a method mark because it shows correct reasoning, not only a guessed formula.

Triangular Numbers and Their Link to Quadratics

Triangular numbers are 1, 3, 6, 10, 15, 21, ... and follow Tn = n(n+1)/2. They represent totals formed by adding consecutive integers: 1, then 1+2, then 1+2+3, and so on. Geometrically, they count points arranged in triangular rows, which is why the name is memorable and useful in problem solving.

Triangular numbers are a special quadratic sequence. Their first differences are 2, 3, 4, 5, ... and their second differences are constant at 1. This means they sit naturally inside the same analysis framework as other quadratic patterns. If you recognise 1, 3, 6, 10 quickly, you can jump straight to n(n+1)/2 and save time.

In exam questions, triangular numbers may appear directly, or they may be embedded in context such as arrangements, seating rows, stacks, handshakes or cumulative totals. When you spot “sum of first n natural numbers,” think triangular formula immediately. That single recognition often turns a long question into a short one.

How the JavaScript Detector Identifies Sequence Type

This page includes a JavaScript calculator that checks your entered terms in a practical order. It tests for Fibonacci recurrence, arithmetic constant difference, geometric constant ratio, triangular number pattern, and quadratic constant second differences. When one rule matches, it displays the type, an nth term formula, and where relevant, sum formulas and convergence information.

Detection is based on numerical consistency, not guesswork. Small tolerance is used for decimal entries, so a sequence like 10, 5, 2.5, 1.25 can still be recognised as geometric with ratio 0.5 even if typed with decimal formats. For any term lookup, you can set n and the calculator returns the nth value using the selected rule. That gives an immediate check against your hand-calculated answer.

If a list does not match a supported pattern, the tool reports that clearly rather than forcing a wrong label. This matters for revision quality: it is better to know a sequence is outside the standard families than to train on incorrect assumptions. Use that feedback to inspect your terms, verify entry order, and re-check whether the question gives a recursive or piecewise rule.

GCSE and A-Level Exam Strategy for Sequence Questions

Sequence marks are often among the most efficient marks on a paper, but only when method is clear. Start each question with a short diagnostic: are you adding a constant amount, multiplying by a constant ratio, using a recurrence rule, or seeing constant second differences? That one decision determines the right formula family. Write that decision down as a short statement. It improves your structure and helps if you need to recover from a later arithmetic slip.

For nth term questions, always show substitution. For sums, write the exact formula first, then replace symbols with numbers, then simplify. Avoid mental jumps that skip lines. Examiners award method marks for visible reasoning, and visible reasoning also protects you from avoidable mistakes. Where calculators are allowed, use them for arithmetic, not for algebra choices. You still need to pick the correct formula yourself.

At A-Level, you may need to combine sequence methods with logs, proof or modelling interpretation. Keep units and interpretation statements where needed. For example, in a growth model, explain what the ratio means in percentage terms. In a decay model, explain why the ratio is between 0 and 1. Small interpretation lines can secure communication marks and make your final answer more defensible.

Practice under timing. Spend less time on detection by learning the visual cues: constant difference means arithmetic, constant ratio means geometric, second difference means quadratic, recursive “sum previous two” means Fibonacci. Fast recognition leaves more time for algebra and checking. After solving, use this calculator to verify nth term outputs and sums before moving on.

Common Mistakes and Fast Self-Checks

Most sequence errors come from form, not theory. The first quick check is index alignment: if your sequence starts at n=1, make sure your nth term does too. The second check is bracket safety: in a+(n-1)d, expand only after the bracket is correct. The third check is condition checking: only use geometric sum to infinity when |r|<1. If you apply these three checks consistently, your accuracy improves immediately.

Another frequent issue is treating arithmetic and geometric formulas as interchangeable. If differences are constant, do not use powers. If ratios are constant, do not use linear n terms. For quadratic patterns, never rely on first differences alone. Confirm second differences explicitly and write that sentence in your working. For triangular numbers, verify first terms quickly against 1, 3, 6, 10 before committing to n(n+1)/2.

Final check before submitting: calculate one known early term from your nth term formula, such as n=1 or n=2, and compare with the original list. This catches sign and constant errors in seconds. A one-line verification can rescue several marks.

Frequently Asked Questions