What is an arithmetic and geometric sequence calculator?
This sequence calculator finds a requested term and the sum of the first n terms when you already know the first term and the sequence rule. Choose arithmetic for a fixed difference between consecutive terms, or geometric for a fixed multiplication ratio. The large answer is the nth term, while a supporting card reports the finite sum. Both values use exact rational arithmetic.
Arithmetic progression and geometric progression are alternative names for the same two families. This page does not guess a rule from an arbitrary list of numbers. A short list can fit many possible rules, so automatically choosing one would not prove what happens next. If you simply want the average of supplied observations, use the Average Calculator; a sequence formula answers a different question.
The mathematics applies equally in school and college work across English-speaking countries. No local currency or measurement system is assumed. The entries may describe a textbook exercise or a simplified pattern, but choosing an appropriate rule remains your responsibility.
How to find an nth term and sequence sum
- Select Arithmetic or Geometric.
- Enter the first term a₁.
- Enter the common difference d or common ratio r.
- Enter n, a whole number from 1 through 1,000.
- Calculate and read the nth term, finite sum and calculation steps.
Indexing starts at term one. Therefore n = 1 returns the first term and its one-term sum. Do not count an extra increment before that first term. In arithmetic mode, the change field is an amount added each time, not a percentage. In geometric mode, it is the multiplier itself: a ten-percent increase uses ratio 1.1, not 10.
You can enter a negative difference or ratio, a zero first term, or fractions such as 1/2. If you need to convert a percentage change into a multiplier, use the Percentage Calculator for a separate arithmetic check. All visible fields must be completed. The form begins empty, editing clears stale answers, and Reset returns it to a blank starting state.
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For a separate arithmetic check, you may browse scientific calculators (paid link). Check the model’s supported functions; basic scientific devices do not necessarily support matrix operations or sequence sums. No purchase is required.
Arithmetic sequence formula and finite sum
An arithmetic sequence adds a fixed difference. Its nth term is aₙ = a₁ + (n − 1)d because there are n − 1 transitions from the first term to the nth term. A negative difference creates a decreasing sequence; a zero difference keeps every term equal to the first term. Fractional differences work the same way as integer differences.
The finite sum can be computed from the first and last terms: Sₙ = n(a₁ + aₙ)/2. The calculator first finds the last term, then substitutes the actual values into that expression. It does not round the average of the endpoints. A second pass builds the sequence by repeated addition and verifies both the last term and total exactly.
For a₁ = 3, d = 4 and n = 5, the terms are 3, 7, 11, 15 and 19. The fifth term is 3 + 4 × 4 = 19, and the sum is 5 × (3 + 19)/2 = 55. Constant absolute changes should not be confused with compound percentage growth.
Geometric sequence formula and special ratios
A geometric sequence multiplies by a fixed ratio, so aₙ = a₁rⁿ⁻¹. When r differs from one, the sum is Sₙ = a₁(1 − rⁿ)/(1 − r). The formula is a finite sum; it does not require an infinite series to converge. A ratio with magnitude greater than one can still produce a perfectly valid finite answer.
When r = 1, every term equals a₁ and the sum is n times a₁. This is handled separately to avoid dividing zero by zero. When r = 0, the first term stays a₁ and every later term is zero. For n = 1, the initial exponent contributes a factor of one in this sequence formula; no undefined general-purpose 0⁰ interpretation is needed.
A negative ratio alternates signs. A zero first term produces all zeros for any supported finite ratio. The Exponent Calculator can check individual powers separately, while this page combines those powers with the sequence term and sum rules.
Worked geometric and fractional sequence examples
For a₁ = 2, r = 3 and n = 4, the terms are 2, 6, 18 and 54. The fourth term is 2 × 3³ = 54, and the finite sum is 2 × (1 − 3⁴)/(1 − 3) = 80. There are three multiplications before the fourth term, not four.
For a₁ = 8, r = 1/2 and n = 4, the terms are 8, 4, 2 and 1. The fourth term is 1 and the finite sum is 15. For a₁ = 2, r = −1 and n = 4, the terms are 2, −2, 2 and −2, so the total is zero. A zero sum does not mean every term is zero.
For an arithmetic sequence beginning at 1/2 with difference 1/3, term four is 3/2 and the four-term sum is 4. You can compare the accepted fractions with the Fraction Calculator. Exact fractions preserve small differences that may be hidden by rounded decimal displays.
Reading the result and calculation explanation
The main result is a single term, not a list of all possible sequence values. The cards distinguish the first term, common difference or ratio, requested index and finite sum. The explanation shows the formulas with your values, followed by up to the first twelve generated terms. If n exceeds twelve, the preview ends with an ellipsis; that does not mean the sum stops at twelve.
A separate recurrence-based calculation verifies all requested terms through n, even when only twelve are previewed. Copy, Download and Print include the result, finite sum and displayed steps. The number of explanation steps follows the calculation path rather than a fixed five-step template.
Supported sequences and exact-size limits
The term number must be a positive integer no greater than 1,000. Number inputs must lie between −10,000 and 10,000. Fractions have numerator and denominator magnitude limits of 10,000; decimals allow at most six places. Scientific notation, symbolic expressions and omitted entries are rejected. Calculations are limited to 512 digits in an exact numerator or denominator; rapidly growing or shrinking geometric sequences can reach that limit before term 1,000.
A size-limit message is not a rounded answer or proof that the sequence is invalid. Reduce the term count or use a suitable arbitrary-precision mathematics package. This page does not solve backwards for n, infer differences from a list, calculate recursive sequences such as Fibonacci, or evaluate infinite-series convergence. It is also not a financial projection tool: real investments may require fees, timing, taxes and changing returns.
The calculator does not save your sequence entries. The website’s separately disclosed analytics and advertising services remain in place. Check the first term, index convention and stated rule in the original exercise before relying on a mathematically correct result.
Arithmetic & Geometric Sequence Calculator FAQs
Sources and methodology
These educational references explain the mathematical ideas. This independently implemented calculator is not affiliated with their publisher. Its supported methods, exact checks and limits are documented above.