Number Sequence Calculator - Arithmetic and Geometric nth Term and Sum

A number sequence calculator answers two school questions for a regular pattern: what is the nth term, and what is the sum of the first n terms? The pattern is either arithmetic (add the same common difference d every step) or geometric (multiply by the same common ratio r every step). Those two families look similar as lists and are completely different as formulas. Mixing them is how 3, 8, 13, 18 gets a geometric “ratio” of 8/3 that was never in the problem.

Arithmetic example, which is the default on this page: a = 3, d = 5, n = 10. The terms are 3, 8, 13, 18, 23, 28, 33, 38, 43, 48. The 10th term is 48. The sum of those ten terms is 255. If this tool shows anything else on those inputs, the closed form was implemented wrong - not a rounding debate.

Geometric cousin of the same first term: a = 3, r = 2, n = 5 is 3, 6, 12, 24, 48. Same 48 at the end, different story: each step doubled instead of adding 5. The nth-term formula is a power, so large n or |r| > 1 can overflow a JavaScript number; this calculator rejects non-finite results instead of printing Infinity as if it were a term.

1. Choose Arithmetic or Geometric.
2. Enter first term a, then d or r, then n as a positive integer from 1 to 10,000.
3. Read a_n, S_n, and - when n ≤ 500 - the first min(n, 30) terms so you can check the pattern by eye.
4. Copy the summary or Clear back to the default 3, 5, 10 checksum. Inputs stay in this browser up to 30 days.

Limits: n must be an integer 1…10,000 for the closed formulas. Term listing is rejected (omitted) when n > 500; at most 30 terms are printed even when listing is allowed. Infinite series are not computed. IEEE-754 overflow is an error, not a result.

a₁₀ = 3 + (10 − 1) × 5 = 3 + 45 = 48. Index 1 is the first term, so you add d nine times to reach term 10, not ten times. Off-by-one on (n − 1) is the most common arithmetic-sequence error: using n instead of n − 1 yields 53, which is a₁₁.

Negative d is a decreasing sequence: a = 20, d = −4, n = 5 is 20, 16, 12, 8, 4. Zero d is a constant sequence; every term equals a. Neither case needs a different formula. Fractions and decimals are allowed; n is the only quantity that must be a whole number (you cannot have the 3.7th term of a discrete sequence on this page).

For a = 3, d = 5, n = 10: S₁₀ = 10/2 × (2 × 3 + 9 × 5) = 5 × (6 + 45) = 5 × 51 = 255. Equivalently first + last = 3 + 48 = 51, times n/2 = 5, same 255. Pairing 3 with 48, 8 with 43, and so on is Gauss's trick: five pairs of 51.

This is a finite sum. An arithmetic series with d ≠ 0 diverges if you try to add forever; there is no infinite-sum button here. If a worksheet asks for an infinite geometric sum, that is a different object and a different formula, and this tool will not pretend n = ∞.

Powers come from the exponent calculator family: rⁿ⁻¹ is an ordinary power. Negative r alternates sign. r = 0 after the first term is all zeros for n ≥ 2. r = 1 is the trap in the sum formula: 1 − r = 0, so the textbook fraction is undefined. Every term equals a, so the sum of n terms is simply n × a. This calculator switches to that identity instead of dividing by zero.

|r| < 1 is the case where an infinite geometric series would converge to a/(1 − r). That infinite sum is not computed here. Enter a finite n. If you wanted the infinite sum, you would still need |r| < 1 and a different tool or a by-hand last step - this page will not quietly replace S_n with S_∞.

Mean of listed terms is not S_n / n for a geometric sequence in any special way beyond the definition of an average; if you copy the listed prefix into the average calculator, remember the prefix may be only 30 terms, not the whole n.

Closed forms are O(1): one multiplication for an arithmetic nth term, a power for a geometric nth term. Listing every term is O(n) and noisy. This page therefore computes formulas for n up to 10,000 and refuses to list when n > 500. When it does list, it shows only the first min(n, 30) terms - enough to see the pattern, not a 500-line paste.

If you need 400 random integers instead of a closed form, that is the random number generator, not a sequence. Sequences are determined; random lists are not.

Switch to geometric mode and try a = 2, r = 3, n = 6. Terms: 2, 6, 18, 54, 162, 486. nth term a₆ = 2 × 3⁵ = 2 × 243 = 486. Finite sum S₆ = 2(1 − 3⁶)/(1 − 3) = 2(1 − 729)/(−2) = 2(−728)/(−2) = 728. Check by adding: 2+6=8, +18=26, +54=80, +162=242, +486=728. If r were 1 instead of 3, those six terms would all be 2 and the sum would be 12, which is 6 × 2, not a broken fraction.

Negative ratio: a = 5, r = −1, n = 4 yields 5, −5, 5, −5. nth term 5 × (−1)³ = −5. Sum 5(1 − (−1)⁴)/(1 − (−1)) = 5(1 − 1)/2 = 0, which matches two pairs that cancel. Alternating geometric sequences are still geometric; they are not arithmetic with a flipping d.

Subtract consecutive terms. If the differences are constant, the sequence is arithmetic and that constant is d. Divide consecutive terms (when they are nonzero). If the ratios are constant, the sequence is geometric and that constant is r. 4, 7, 10, 13 has differences 3, 3, 3 → arithmetic, a = 4, d = 3. 4, 8, 16, 32 has ratios 2, 2, 2 → geometric, a = 4, r = 2. 4, 8, 14, 22 has differences 4, 6, 8 - neither family; this calculator will still compute an a, d, n if you force arithmetic, but it will not match the third term. Garbage in is a well-defined wrong sequence, not a detection algorithm.

Quadratic sequences (square numbers, triangular numbers) change by an arithmetic sequence of differences. They are not modes on this page. Recursive definitions such as Fibonacci (each term the sum of two previous) are also out of scope: there is no single d or r. Use the closed arithmetic or geometric formulas only when the problem states common difference or common ratio, or when the first terms pass the subtract-or-divide test.

Word problems hide the parameters. “A theater has 12 seats in row 1 and 3 extra seats each row after” is arithmetic a = 12, d = 3; n is the row number; the nth term is seats in that row; S_n is seats in the first n rows. “A culture triples every period starting at 200 cells” is geometric a = 200, r = 3. Translating English into a, d or r, and n is the actual exam skill; the calculator only does the arithmetic after that translation. If a problem gives two terms and asks for d, subtract them and divide by how many steps apart they are: a_7 − a_3 = 4d, so d = (a_7 − a_3)/4. Then type a, d, n here. For geometric, divide: a_7 / a_3 = r^4, so r is the fourth root, which you can check with an exponent tool, then paste r into this page. The calculator will not solve for d or r from two arbitrary terms; it is an nth-term and sum engine once the parameters are known.

For the default arithmetic list, S₁₀ / 10 = 255 / 10 = 25.5, which happens to equal (first + last) / 2 = (3 + 48) / 2. That 25.5 is the average of the ten terms, not the 10th term 48, and not the first term 3. Students who divide the sum by n and call it a_n fail the checksum on this page immediately. Geometric sequences do not even enjoy a simple “average equals midpoint of first and last” identity in the same way: for 2, 6, 18, 54, 162, 486 the mean is 728/6 ≈ 121.333, which is not (2+486)/2.

Partial sums are another homework object. S₄ for a=3, d=5 is 4/2 × (6 + 15) = 2 × 21 = 42, matching 3+8+13+18. This calculator always uses the n you typed as both the index of a_n and the number of terms in S_n. There is no separate “sum from term 5 to term 12” mode. To get a partial range, compute S₁₂ − S₄ by hand (sum of first 12 minus sum of first 4 leaves terms 5 through 12). That identity works for both arithmetic and geometric finite sums.

Large geometric n with |r| > 1 will hit JavaScript Infinity. 2 × 10⁴⁰⁰ is not a term this engine can store. The error message tells you to shrink |r| or n; it does not print Infinity as if it were 486. Arithmetic sequences overflow much later because they grow linearly: a + (10000−1)d stays finite for ordinary school d. That is an implementation limit sitting on top of the algebra, disclosed on purpose.

Recurrence form a_n = a_n−1 + d (or × r) is equivalent to the closed form when you start at a_1 = a. Indexing wars (some books call the first term a_0) would shift every formula by one. This page uses a_1 = a, a_n = a + (n−1)d, which is the usual high-school convention. If your lecturer starts at n = 0, subtract one from their n before you type it here, or rewrite their a_0 as the first term a.

Take a = 1, r = 1/2, n = 4. Terms: 1, 0.5, 0.25, 0.125. nth term a₄ = (1/2)³ = 0.125. Finite sum S₄ = 1(1 − (1/2)⁴)/(1 − 1/2) = (1 − 1/16)/(1/2) = (15/16)×2 = 1.875. The infinite sum everyone memorizes is a/(1 − r) = 1/(1/2) = 2. This calculator reports 1.875, not 2. The missing 0.125 is all remaining terms after n = 4. If a worksheet wanted 2, it asked for S_∞, which is explicitly out of scope. Type a larger n to watch S_n approach 2; do not expect the page to jump to the limit.

The same honesty applies to |r| ≥ 1: there is no finite infinite-sum, so refusing S_∞ is not a missing feature, it is the definition of divergence. Arithmetic series with d ≠ 0 also diverge; Gauss's pairing trick only packages a finite n. Keep n in 1…10,000, keep listing honest (no list above 500, only 30 terms shown), and treat overflow as an error. That is the whole contract of this sequence tool. Privacy is the same contract as the rest of DevOkk scientific calculators: a, d, r, and n never leave this browser, and a 30-day localStorage draft is only on this device.