Sequence Calculator
Find the nth term and the running sum of an arithmetic or geometric sequence from the first term, common difference or ratio, and number of terms.
Sequence Calculator
Calculator
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Formula: Arithmetic: aₙ = a₁ + (n-1)d, Sₙ = n(a₁+aₙ)/2 | Geometric: aₙ = a₁ × rⁿ⁻¹, Sₙ = a₁(1-rⁿ)/(1-r)
Worked example — a₁₀ = 48, Sum = 255
Formula
Arithmetic: aₙ = a₁ + (n-1)d, Sₙ = n(a₁+aₙ)/2 | Geometric: aₙ = a₁ × rⁿ⁻¹, Sₙ = a₁(1-rⁿ)/(1-r)
Arithmetic sequences add a constant difference. Geometric sequences multiply by a constant ratio.
Worked Examples
Example 1: Arithmetic
Problem:a₁=3, d=5, n=10
Solution:a₁₀ = 3+9×5 = 48, Sum = 10×(3+48)/2 = 255
Result:a₁₀ = 48, Sum = 255
Frequently Asked Questions
What is the difference between arithmetic and geometric sequences?
Arithmetic adds a constant (2,4,6,8...). Geometric multiplies by a constant (2,4,8,16...). Geometric sequences grow much faster.
How do I find any term in an arithmetic sequence without listing all terms?
Use the formula aₙ = a₁ + (n - 1)d, where a₁ is the first term, d is the common difference, and n is the term number. For example, in the sequence 5, 8, 11, 14... with d=3, the 20th term is 5 + (20-1)×3 = 5 + 57 = 62, without needing to list all 20 terms.
When does a geometric sequence diverge vs converge?
A geometric sequence with common ratio |r| < 1 converges to zero (terms shrink toward 0). With |r| > 1, the sequence diverges (terms grow without bound). With r = 1, all terms equal a₁. With r = -1, the sequence alternates between a₁ and -a₁. The infinite geometric series sum exists only when |r| < 1 and equals a₁ / (1 - r).
What are real-world examples of arithmetic and geometric sequences?
Arithmetic sequences appear in depreciation (fixed amount per year), savings with fixed contributions, and equally-spaced measurements. Geometric sequences model population growth, compound interest, radioactive decay, and virus spread. The key distinction is whether change is additive (arithmetic) or multiplicative (geometric).
How do I find the common difference or ratio from two known terms?
For an arithmetic sequence, d = (aₘ − aₙ)/(m − n). If a₃ = 11 and a₇ = 27, then d = (27 − 11)/(7 − 3) = 4. For a geometric sequence, r = (aₘ/aₙ) raised to the power 1/(m − n). If a₂ = 6 and a₅ = 48, then r is the cube root of 8, which is 2. Once d or r is known, back-solve for a₁ with the explicit formula.
Why does the geometric sum formula break down when the ratio equals 1?
Sₙ = a₁(1 − rⁿ)/(1 − r) carries 1 − r in its denominator, which is zero at r = 1. That is a failure of the formula, not of the sequence: when r = 1 every term equals a₁, so the sum is simply n × a₁. Calculators must special-case it, and results also become numerically delicate when r is extremely close to 1.
Can the common difference or the common ratio be negative?
Yes. A negative common difference gives a strictly decreasing arithmetic sequence such as 20, 17, 14, 11 with d = −3. A negative common ratio makes a geometric sequence alternate in sign, such as 3, −6, 12, −24 with r = −2. Whether a geometric sequence shrinks toward zero depends on the absolute value of r, not its sign, so r = −0.5 still converges.
When should I use a recursive definition rather than an explicit formula?
A recursive definition states each term in relation to the one before it, aₙ = aₙ₋₁ + d for arithmetic or aₙ = r × aₙ₋₁ for geometric, plus a starting value. An explicit or closed form gives aₙ straight from n, such as aₙ = a₁ + (n − 1)d. Recursion suits step-by-step modelling; the closed form is what you want when jumping directly to the 500th term.
How many terms does an arithmetic sequence have if I know its last term?
Rearrange aₙ = a₁ + (n − 1)d into n = (aₙ − a₁)/d + 1. For a sequence starting at 7 with d = 4 and ending at 71, n = (71 − 7)/4 + 1 = 16 + 1 = 17 terms. If that division does not come out as a whole number, the value you were given is not actually a term of the sequence.
What is the difference between a sequence and a series?
A sequence is the ordered list of terms itself; a series is what you get by adding those terms together. Sequence Calculator reports both: aₙ is the nth term of the sequence, while Sₙ is the sum of the first n terms, the nth partial sum of the series. A sequence can converge while its series diverges, as the terms 1/n do.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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