🔢 Number Sequence Calculator

1. Enter at least 3 terms of your sequence, separated by commas.
2. For example, enter 2, 6, 18, 54 for a geometric sequence.
3. Click Calculate to identify the pattern.
4. See the next 5 terms, the nth term formula and the partial sum.
5. Try different sequences to compare arithmetic and geometric patterns.

What if my sequence is neither arithmetic nor geometric?

Some sequences follow other patterns: Fibonacci (each term is the sum of the two preceding), quadratic (the second differences are constant) or alternating. The calculator will report that it cannot identify the sequence as arithmetic or geometric. Try entering more terms or look for a different pattern manually.

What is the difference between arithmetic and geometric?

In an arithmetic sequence, you add a fixed number to get the next term (2, 5, 8, 11 with d = 3). In a geometric sequence, you multiply by a fixed number (2, 6, 18, 54 with r = 3). Arithmetic grows linearly; geometric grows exponentially. Over many terms, a geometric sequence with r > 1 quickly outpaces an arithmetic sequence.

How many terms do I need to enter?

At least 3 terms are required to detect a pattern. Entering more terms improves accuracy: 3 terms uniquely determine an arithmetic or geometric sequence, but 4 or more terms help confirm that the pattern is consistent and not a coincidence.

What is the sum formula used for?

The partial sum Sₙ gives the total of the first n terms. For arithmetic sequences, it is useful for calculating total payments in a payment plan with fixed increases. For geometric sequences, it is used in compound interest (summing returns over time), annuities and loan amortization. The geometric sum formula is the basis for calculating the present value of a series of future cash flows.

When identifying a sequence pattern, always check both arithmetic (constant difference) and geometric (constant ratio) possibilities. For arithmetic sequences, subtract consecutive terms: if all differences are the same, d is confirmed. For geometric sequences, divide consecutive terms: if all ratios are the same, r is confirmed. Watch out for sequences that are both arithmetic and geometric: a constant sequence like 5, 5, 5, 5 is technically both (d = 0 and r = 1). In real-world applications, arithmetic sequences model linear growth (simple interest, fixed-amount savings plans), while geometric sequences model exponential growth (compound interest, population models, viral spread). When using the sum formula for geometric sequences with r between −1 and 1, the infinite sum converges to a₁ / (1−r), which is the basis for present value calculations in finance. Always verify your pattern by checking that the predicted next term matches what you would expect from the rule.