Sequence Calculator
Choose a sequence type, enter your first term and common difference or ratio, then specify which term you want. The calculator returns the nth term, the partial sum up to n, and lists up to 20 terms. Step-by-step work shows exactly how each result was reached.
Formula
Worked example
Arithmetic example: first term 2, common difference 3, n = 10. The 10th term = 2 + (10-1) x 3 = 2 + 27 = 29. Sum of first 10 terms = 10 x (2 + 29) / 2 = 155. Geometric example: first term 3, ratio 2, n = 8. The 8th term = 3 x 2^7 = 384. Sum = 3 x (1 - 2^8)/(1 - 2) = 765.
What is a sequence?
A sequence is an ordered list of numbers following a definite rule. Each number in the list is called a term. The three most common types you will encounter in school and everyday life are arithmetic sequences, geometric sequences, and the Fibonacci sequence. An arithmetic sequence adds the same number (the common difference) at every step, so the terms change at a constant rate. A geometric sequence multiplies by the same number (the common ratio) at every step, so the terms grow or shrink exponentially. In the Fibonacci sequence each term is the sum of the two terms before it, producing the famous spiral pattern found throughout nature.
How to find the nth term and the partial sum
For an arithmetic sequence with first term a1 and common difference d, the nth term is a(n) = a1 + (n-1) x d, and the sum of the first n terms is S(n) = n x (a1 + a(n)) / 2. For a geometric sequence with first term a1 and common ratio r, the nth term is a(n) = a1 x r^(n-1), and the partial sum is S(n) = a1 x (1 - r^n) / (1 - r) when r is not equal to 1. The Fibonacci sequence starts 1, 1, 2, 3, 5, 8, ... and the sum of the first n Fibonacci numbers equals F(n+2) - 1, where F is the standard Fibonacci numbering starting at F(1) = 1.
Arithmetic vs. geometric sequences
Arithmetic sequences change by addition, so their graphs form a straight line. Geometric sequences change by multiplication, so their graphs are exponential curves. When |r| is greater than 1 the geometric sequence diverges rapidly; when |r| is less than 1 each term shrinks toward zero and the infinite sum converges to a1 / (1 - r). Knowing which type of sequence you have determines which formula to use. The key test: subtract consecutive terms - if the difference is constant it is arithmetic; divide consecutive terms - if the quotient is constant it is geometric.
Real-world uses of sequences
Sequences appear across mathematics, science, finance, and computing. Arithmetic sequences describe constant-rate growth such as a salary that increases by the same amount each year, or equal monthly deposits into a savings account. Geometric sequences model compound interest, population growth, radioactive decay, and the spread of a signal in a network. Fibonacci numbers appear in the spiral arrangement of sunflower seeds, the branching of trees, the structure of galaxies, and many optimisation algorithms in computer science. Understanding sequences is a foundation for calculus, where infinite series converge to functions such as e^x and sin(x).
Common sequence formulas at a glance
| Sequence type | nth term | Partial sum S(n) | Infinite sum |
|---|---|---|---|
| Arithmetic | a1 + (n-1)d | n(a1 + an) / 2 | Diverges |
| Geometric (|r|<1) | a1 x r^(n-1) | a1(1-r^n)/(1-r) | a1 / (1-r) |
| Geometric (|r|>1) | a1 x r^(n-1) | a1(r^n-1)/(r-1) | Diverges |
| Fibonacci | F(n-1)+F(n-2) | F(n+2) - 1 | Diverges |
n = term number, a1 = first term, d = common difference, r = common ratio.
Frequently asked questions
What is the difference between arithmetic and geometric sequences?
An arithmetic sequence has a constant difference between consecutive terms (you add or subtract the same number each step), while a geometric sequence has a constant ratio between consecutive terms (you multiply by the same number each step). For example, 2, 5, 8, 11 is arithmetic with d = 3, while 3, 6, 12, 24 is geometric with r = 2.
How do I find the nth term of a geometric sequence?
Multiply the first term by the common ratio raised to the power (n - 1). The formula is a(n) = a1 x r^(n-1). For a first term of 3 and ratio of 2, the 8th term is 3 x 2^7 = 3 x 128 = 384.
What is the sum of an infinite geometric series?
When the absolute value of the common ratio is less than 1, the terms shrink toward zero and the series converges. The infinite sum is a1 / (1 - r). For example, the series 1 + 0.5 + 0.25 + 0.125 + ... converges to 1 / (1 - 0.5) = 2. If |r| is 1 or greater the series diverges (the sum grows without bound).
How is the sum of the first n Fibonacci numbers calculated?
The sum of the first n Fibonacci numbers equals F(n+2) - 1, where F(1) = 1, F(2) = 1. For example, the sum of the first 5 Fibonacci numbers (1+1+2+3+5 = 12) equals F(7) - 1 = 13 - 1 = 12.
Can the common ratio of a geometric sequence be negative?
Yes. A negative common ratio causes the terms to alternate between positive and negative values. For example, first term 1 and ratio -2 gives: 1, -2, 4, -8, 16, ... The formulas for the nth term and partial sum still work correctly with a negative ratio.
What is the formula for the sum of an arithmetic sequence?
The sum of the first n terms is S(n) = n x (a1 + a(n)) / 2, which is the number of terms multiplied by the average of the first and last term. Equivalently, S(n) = n x (2 x a1 + (n - 1) x d) / 2. For 10 terms starting at 2 with difference 3, S(10) = 10 x (2 + 29) / 2 = 155.