Number Sequences Explained: Arithmetic, Geometric and the Sums That Follow

What a number sequence actually is

A number sequence is an ordered list of numbers generated by a rule. The rule is what makes a sequence different from a random list: knowing the first term and the rule lets you write down the second, third, hundredth, or millionth term without ever seeing the ones in between. Two patterns cover the vast majority of cases you meet in school, exam papers, and applied work — the arithmetic sequence, where each step adds the same amount, and the geometric sequence, where each step multiplies by the same factor. The number sequence calculator at the top of this page handles both: enter the first term, the common difference or ratio, and how many terms you want, and it returns the nth term, the partial sum, the full list of values, and — for geometric sequences with a ratio between −1 and 1 — the infinite sum the series converges to.

The distinction matters because the two families behave very differently. An arithmetic sequence grows like a straight line, so doubling the number of terms roughly doubles the final value. A geometric sequence grows like an exponential curve, so doubling the number of terms can square the final value. The famous chess-board grain story — one grain on the first square, two on the second, four on the third, doubling each time — is a geometric sequence with ratio 2. Sum it across the 64 squares and you get 18,446,744,073,709,551,615 grains, roughly a thousand years of modern global wheat production. The same arithmetic sequence growing by one grain per square would top out at 2,080. The rule choice changes everything.

The formulas behind each sequence type

The arithmetic sequence rule is:

a_n = a_1 + (n − 1) d

Read aloud: the nth term equals the first term plus n minus one common differences. The minus-one is the part that trips people up — the first term is already a₁, so to reach the second term you add d once, not twice. Indexing from one (rather than zero) is the convention textbooks use and the one the number sequence calculator follows. The partial sum of the first n terms is:

S_n = n/2 × (a_1 + a_n)

This is the formula Gauss is supposed to have rediscovered at age seven when his schoolmaster asked the class to add the integers from 1 to 100. Pair the smallest with the largest, the second-smallest with the second-largest, and so on — every pair sums to 101, there are 50 pairs, total 5,050. The pattern generalises: an arithmetic sequence is symmetric around its midpoint, so the average of the first and last term equals the average of every term, and the sum is that average times the count.

The geometric sequence rule is:

a_n = a_1 × r^(n − 1)

Same convention: the first term is a₁, and to reach the second you multiply by r once. The partial sum has two cases. When r is anything other than 1:

S_n = a_1 × (1 − r^n) / (1 − r)

When r equals exactly 1, every term equals a₁ and the formula above collapses to 0 over 0, so the sum is simply n × a₁. The calculator handles the special case to avoid the division. The infinite sum — the limit as n goes to infinity — only exists when the absolute value of r is less than 1:

S_∞ = a_1 / (1 − r)   (only when |r| < 1)

That single line is the result behind Zeno's paradox of Achilles and the tortoise. If Achilles halves the gap with each step (geometric, r = 1/2), the infinite series of step lengths sums to a finite distance — twice the initial gap — which is why he overtakes the tortoise in finite time despite needing infinitely many steps. When |r| ≥ 1 the partial sums grow without bound (or oscillate without settling), and the infinite sum is said to diverge. The number sequence calculator reports the infinite sum for convergent geometric series and labels divergent ones as such.

Worked example: a savings plan in two forms

Imagine you set aside £50 in week one, and you want to know what happens under two different escalation rules: adding £10 each week (arithmetic) or multiplying the deposit by 1.05 each week (geometric). After 20 weeks, which ends up with more in the pot?

Arithmetic case. First term 50, common difference 10, 20 terms. The 20th deposit is:

a_20 = 50 + (20 − 1) × 10 = 50 + 190 = 240

The total deposited over the 20 weeks is:

S_20 = 20/2 × (50 + 240) = 10 × 290 = 2,900

You end up depositing £2,900, with the final week's contribution at £240.

Geometric case. First term 50, common ratio 1.05, 20 terms. The 20th deposit is:

a_20 = 50 × 1.05^19 ≈ 50 × 2.5270 ≈ 126.35

The total deposited is:

S_20 = 50 × (1 − 1.05^20) / (1 − 1.05) ≈ 50 × (1 − 2.6533) / (−0.05) ≈ 1,653

Geometric escalation at 5% a week ends with a single contribution of about £126 and a cumulative deposit of about £1,653 — roughly £1,250 less than the arithmetic case. The arithmetic plan wins because 10 a week is a steeper absolute escalator at this scale than 5% compounding off a small base. The crossover happens around week 60: by then the geometric series has compounded enough to outpace the linear escalator, and from there it pulls ahead fast. That crossover behaviour is the practical reason engineers, savers, and scientists need to know which kind of growth they are dealing with — linear plans look better in the short run and lose badly in the long run, exponential plans look weak early and dominate eventually. Drop these numbers into the number sequence calculator directly to verify, then try 60 terms instead of 20 to see the crossover.

Factors that change a sequence's behaviour

The sign of the common difference or ratio

A negative common difference produces a decreasing arithmetic sequence: 100, 90, 80, 70, … is the classic countdown pattern. The formulas are unchanged — the calculator simply returns smaller (and eventually negative) terms as n grows. A negative common ratio produces an alternating geometric sequence: 3, −6, 12, −24, … flips sign on every step. The absolute value of the terms still grows (or shrinks) at the rate |r|, but the partial sums oscillate. When r is between −1 and 0, the infinite sum still converges, just to a value smaller than the first term because the alternating signs cancel.

Whether |r| is bigger or smaller than 1

This single inequality decides whether a geometric sequence grows or shrinks without bound. When |r| > 1, the terms get larger forever and the partial sums diverge. When |r| < 1, the terms shrink toward zero and the partial sums converge to a finite limit. When |r| = 1 exactly, the sequence is either constant (r = 1) or alternates between a₁ and −a₁ (r = −1), and the sum either grows linearly or fails to settle. The boundary is so important that it has a name in mathematical analysis — the radius of convergence — and it governs everything from compound-interest formulas to perturbation theory in physics.

The scale of the first term

a₁ acts as a multiplier on every other term — double the first term and every subsequent term doubles too. That sounds trivial, but it matters when you are modelling a process where the starting value is itself uncertain. A 10% uncertainty in a₁ propagates straight through to a 10% uncertainty in a₁₀₀, no matter how many terms you generate. The shape of the curve does not change; only the height does.

Whether you index from 0 or from 1

Mathematicians and most calculus textbooks index from 1, so a₁ is the first term and a₂ is the second. Computer scientists and most programming languages index from 0, so a₀ is the first element. The two conventions differ by one in every nth-term formula. If a textbook says "the 10th term of the Fibonacci sequence is 55" it means F₁₀ = 55 with F₁ = 1 and F₂ = 1; a programmer writing fib[10] with zero-indexing gets 89 instead. The number sequence calculator uses the textbook convention: the count you enter is the number of terms, and the nth term is the last one in the list.

Floating-point overflow at high n with large |r|

Geometric sequences grow fast. With r = 10 and n = 309, the nth term hits about 10³⁰⁸, which is the edge of double-precision floating point — beyond that, JavaScript (and most other languages) returns Infinity. The calculator caps n at 1000 to keep results sensible, but you can hit overflow well before that with a large ratio. If you need exact arithmetic on huge geometric sequences, you want a symbolic system such as SymPy or Mathematica, not a floating-point one.

How to spot which kind of sequence you have

• Compute consecutive differences first. If the differences are constant, it is arithmetic and the common difference is whatever value you see. If the differences themselves grow, it is probably geometric or something more exotic.
• Then compute consecutive ratios. If those are constant, you have a geometric sequence and the common ratio is the constant value. Mixed behaviour suggests a quadratic, Fibonacci-style recurrence, or a more unusual rule that needs case-by-case analysis.
• Plot the first ten terms on a log axis. A geometric sequence plots as a straight line on a log-y axis; an arithmetic sequence curves on a log axis. This is the fastest visual diagnostic for messy data.
• Beware of look-alike patterns. 1, 4, 9, 16, 25 looks like it could be geometric, but the ratios — 4, 2.25, 1.78, 1.56 — are not constant. These are perfect squares (n²), a polynomial sequence, not a geometric one.
• Use exact starting values when possible. Tiny rounding in the first term compounds through a geometric series. If the first term is really 1/3, type 0.3333333333 — not 0.33.
• Sanity-check the sum against the formula. For arithmetic sequences the average of first and last term times the count should match. For convergent geometric series, the partial sum should approach a₁/(1−r) as n grows. Mismatches usually mean an indexing-by-one error.

Common mistakes

Off-by-one on the nth-term formula. The exponent is n − 1, not n. Writing a₁₀ = a₁ × r¹⁰ instead of a₁ × r⁹ stretches the sequence by one extra multiplication and shifts every result. The same thing happens to beginners with arithmetic — writing a₁₀ = a₁ + 10d instead of a₁ + 9d. The rule of thumb: to reach the nth term, you take n − 1 steps from the first.

Confusing the partial sum with the nth term. The nth term is a single value; the partial sum is the total of the first n terms. For geometric sequences with |r| > 1, the partial sum is dominated by the last term — the sum can easily be 90% the size of the last term alone, which feels wrong until you remember that the last term is bigger than all the previous ones combined.

Forgetting the |r| < 1 condition for infinite sums. A common exam error is writing S∞ = a₁ / (1 − r) for any geometric series. The formula only applies when the series converges. Asking for the infinite sum of 1, 2, 4, 8, 16, … is meaningless — the series diverges to infinity, and the "answer" 1 / (1 − 2) = −1 is a famous false result that turns up in popular maths videos and is technically only valid in a generalised analytic-continuation sense, not as an ordinary sum.

Mixing up "sequence" and "series". A sequence is the list of terms (2, 5, 8, 11, …). A series is what you get when you add them up (2 + 5 + 8 + 11 + …). The two share a generating rule but answer different questions. Exam questions that say "find the value of the series" want the sum; ones that say "find the nth term of the sequence" want a single number.

Treating Fibonacci as geometric. The Fibonacci sequence — 1, 1, 2, 3, 5, 8, 13, 21, … — has approximately constant ratios for large n (the golden ratio, about 1.618), so it looks geometric in the tail. It is not. Fibonacci is defined by a recurrence (each term is the sum of the previous two) and needs a different formula family. Plugging it into a geometric model breaks down for small n. A dedicated Fibonacci tool is the right home for that pattern.

When you need more than this calculator

Number sequences cover arithmetic and geometric patterns cleanly, but real problems sometimes need richer machinery. Recurrence relations (each term defined in terms of one or more previous terms) — Fibonacci, Lucas numbers, Catalan numbers — need their own solvers. Polynomial sequences (squares, cubes, triangular numbers) are best computed directly from the closed form rather than via a difference rule. For continuously compounded growth, the right tool is a compound-interest formula, not a discrete geometric series. For statistical analysis of a sequence of measurements — finding the mean, the standard deviation, or a trend line — see the average calculator. Sequences and series are a foundation, but they are not the whole of discrete mathematics, and knowing when to reach for something else is part of using them well.

Frequently asked questions

What is the difference between a sequence and a series?

A sequence is an ordered list of terms — for example 3, 6, 9, 12, 15. A series is the result of summing the terms of a sequence — for example 3 + 6 + 9 + 12 + 15 = 45. The same rule generates both, but they answer different questions: a sequence asks "what is the nth value?" and a series asks "what is the running total?". The calculator shows both so you do not have to compute them separately.

How do I find the next term if I only know a few values?

For arithmetic sequences, subtract consecutive terms — the common difference is the same gap repeated. For geometric, divide consecutive terms — the common ratio is the same factor repeated. Then plug a₁, the difference or ratio, and the term number you want into the calculator. If consecutive differences and consecutive ratios are both non-constant, the sequence is not a simple arithmetic or geometric one — it might be polynomial, recursive, or defined by an external rule that the data alone cannot reveal.

Can the first term be negative or zero?

Yes for both. A negative first term in an arithmetic sequence just shifts every value down by the same amount — −5, −2, 1, 4, 7 is a perfectly valid arithmetic sequence with d = 3. A first term of zero in a geometric sequence produces a trivial all-zeros sequence, because every term is zero times something. A first term of zero in an arithmetic sequence is fine — 0, d, 2d, 3d, … is just a list of multiples of d.

What does it mean when a geometric series "converges"?

Convergence means the partial sums approach a fixed finite value as the number of terms grows. For a geometric series with |r| < 1, each new term is smaller than the previous one in absolute value, and the sequence of partial sums settles toward a₁ / (1 − r). The classic example: 1 + 1/2 + 1/4 + 1/8 + … converges to 2, no matter how many terms you add. When |r| ≥ 1, the terms stop shrinking, partial sums never settle, and the series is said to diverge.

How are arithmetic sequences used in real life?

Anywhere a quantity grows or shrinks by a fixed step. Loan amortisation schedules with a fixed monthly principal payment, theatre seat numbering with a fixed extra seat per row, salary scales with fixed annual increments, depreciation on a straight-line basis — all arithmetic. The flat shape of the growth curve is what makes them tractable to plan against, because a straight line is the easiest curve to extrapolate.

How are geometric sequences used in real life?

Compound interest is the textbook case — money multiplies by (1 + rate) each period. Population growth in unconstrained conditions, radioactive decay (ratio less than 1), digital signal amplification at constant gain, and bit depth in binary computing (every extra bit doubles the addressable space) all follow geometric rules. The compound shape — slow start, explosive middle, often a constraint that bends the curve later — is the signature.

What is the relationship between a geometric sequence and compound interest?

A balance compounded annually at rate r forms a geometric sequence with common ratio 1 + r. Starting with £1000 at 5% annual interest, the sequence of balances is 1000, 1050, 1102.50, 1157.63, … — geometric with a₁ = 1000 and ratio 1.05. The same formulas apply: the balance after n years is 1000 × 1.05ⁿ, which is the nth term of a geometric sequence (with the zero-indexing convention this time, because year zero is the starting balance). See the compound interest calculator for the dedicated tool.

Why does my answer differ from a textbook by one term?

Almost always an indexing issue. Some textbooks use a₀ as the first term and write the formula as a_n = a_0 × r^n; others use a₁ and write a_n = a_1 × r^(n−1). Both are correct but they describe the same sequence with different labels. If you and the textbook agree on the list of values, you have the right answer — the disagreement is only about which subscript to attach to which value. The calculator uses a₁ as the first term.

Related calculators

• Exponent calculator — powers, roots and reciprocals; the operation underneath every geometric-sequence term.
• Average calculator — mean, median, mode and range; useful when you have a sequence of measurements and want a summary statistic.
• Factor calculator — all factors and prime factorisation of an integer; pairs with sequence work in number theory exercises.
• Compound interest calculator — the finance application of a geometric sequence, with currency formatting and time-period flexibility.
• Logarithm calculator — the inverse of the exponential, useful for solving "how many terms until the sequence reaches X?" problems.