Sequence Calculator
Find the nth term and the sum of an arithmetic or geometric sequence from the first term, the common difference or ratio, and the number of terms.
Cómo usar esta calculadora
- 1Choose an arithmetic (adding) or geometric (multiplying) sequence.
- 2Enter the first term.
- 3Enter the common difference (arithmetic) or common ratio (geometric).
- 4Enter how many terms, and read the nth term and the running sum.
Cómo funciona
Arithmetic and geometric sequences
arithmetic: aₙ = a₁ + (n − 1)d, sum = n/2 · (a₁ + aₙ) geometric: aₙ = a₁ · r^(n − 1) geometric sum = a₁ · (rⁿ − 1) ÷ (r − 1), r ≠ 1 infinite geometric sum = a₁ ÷ (1 − r), |r| < 1
The two most important sequences differ in how each term is built from the last. In an arithmetic sequence you add a fixed common difference every step, producing a straight-line pattern like 2, 5, 8, 11. The nth term is the first term plus (n − 1) differences, and the sum of the first n terms is neatly the average of the first and last term times how many there are. In a geometric sequence you multiply by a fixed common ratio every step, producing exponential growth or decay like 2, 6, 18, 54. Its nth term multiplies the first by the ratio raised to (n − 1), and its sum follows a compact formula involving the ratio raised to the nth power. When the ratio's magnitude is below one, the terms shrink toward zero fast enough that even an infinite sum settles on a finite value.
Ejemplo resuelto
The arithmetic sequence starting at 2 with a common difference of 3 — 2, 5, 8, 11, … — has a 10th term of 2 + 9×3 = 29, and its first ten terms sum to 10/2 × (2 + 29) = 155. A geometric sequence starting at 2 with ratio 3 would instead reach 2 × 3⁹ = 39,366 by its 10th term.
Sequence Calculator: la guía completa
Adding versus multiplying: the two growth patterns
Arithmetic and geometric sequences represent the two fundamental ways a quantity can change step by step: by a fixed amount or by a fixed factor. Arithmetic growth is linear — save $100 a month and your total climbs in equal increments, tracing a straight line. Geometric growth is exponential — earn a fixed percentage return and your money multiplies, tracing a curve that starts gently and then soars. The difference between adding and multiplying at each step is the difference between a modest, predictable rise and the runaway acceleration that makes compound interest so powerful.
This is why the distinction matters far beyond the classroom. Many real quantities that look like steady growth are actually one or the other, and confusing them leads to badly wrong intuitions. People routinely underestimate geometric growth because their instinct is arithmetic: they expect a doubling quantity to grow steadily when it is in fact exploding. Recognising whether a process adds or multiplies each period is the first step to reasoning about it correctly, whether it is savings, population, a rumour spreading, or a virus.
The clever tricks behind the sum formulas
The formula for summing an arithmetic sequence has a famous origin story: the young mathematician Carl Friedrich Gauss, asked to add the numbers 1 to 100, saw that pairing the first with the last (1 + 100), the second with the second-last (2 + 99), and so on gives fifty pairs each summing to 101 — an instant answer of 5,050. That insight generalises to the formula 'number of terms times the average of the first and last', which is all the arithmetic sum formula really says. It works because the terms are evenly spaced, so their average is just the midpoint of the ends.
The geometric sum uses a different but equally elegant trick. Multiply the whole sum by the common ratio and subtract the original sum: almost every term cancels, collapsing a long addition into the compact expression a₁(rⁿ − 1)/(r − 1). This telescoping cancellation is why you never have to add the terms one by one. Both formulas turn what could be a tedious summation of hundreds of terms into a single quick calculation, which is exactly what makes sequences practical rather than merely theoretical.
When an infinite sum is finite
One of the more counterintuitive results in mathematics is that adding infinitely many positive numbers can produce a finite total. This happens for a geometric sequence whose common ratio has magnitude less than one, so each term is a fraction of the last and the terms shrink toward zero quickly. The sum 1 + ½ + ¼ + ⅛ + … never exceeds 2, no matter how many terms you add; it converges to exactly 2. The infinite geometric sum formula, a₁ ÷ (1 − r), captures this limiting value.
This idea of a convergent infinite series is foundational to calculus and shows up throughout science and finance. The present value of a perpetual stream of payments, the resolution of Zeno's ancient paradoxes about motion, the way a bouncing ball's total travel distance is finite despite infinitely many bounces — all rest on convergent geometric series. When the ratio's magnitude is one or more, by contrast, the terms do not shrink and the sum grows without bound. The boundary at |r| = 1 between convergence and divergence is one of the cleanest and most consequential dividing lines in all of mathematics.
Preguntas frecuentes
What is the difference between arithmetic and geometric sequences?
An arithmetic sequence adds a fixed common difference each step (2, 5, 8, 11), growing linearly. A geometric sequence multiplies by a fixed common ratio each step (2, 6, 18, 54), growing exponentially. Adding gives steady growth; multiplying gives accelerating growth or decay.
How do I find the nth term of a sequence?
For an arithmetic sequence, aₙ = a₁ + (n − 1)d, adding the common difference (n − 1) times. For a geometric sequence, aₙ = a₁ · r^(n − 1), multiplying by the common ratio (n − 1) times. So the 10th arithmetic term of 2 with difference 3 is 2 + 9×3 = 29.
How do I sum a sequence quickly?
For an arithmetic sequence, multiply the number of terms by the average of the first and last term: n/2 × (a₁ + aₙ). For a geometric sequence, use a₁(rⁿ − 1)/(r − 1). Both avoid adding the terms one by one, which is what makes long sums instant.
Can an infinite sequence have a finite sum?
Yes, a geometric sequence can, if its common ratio has magnitude less than one. The terms shrink toward zero fast enough that the total converges to a₁ ÷ (1 − r). For example, 1 + ½ + ¼ + … converges to 2. This is a cornerstone idea in calculus.