Number Sequence and Fibonacci Calculator
Generate an arithmetic or geometric sequence from its first term, its common difference or ratio and the number of terms. The calculator lists every term, then shows the nth term, which is the last one, and the sum of the series. Pick Fibonacci to build a sequence where each term is the sum of the two before it, from any two starting terms and for up to 500 terms.
How to use the Number Sequence Calculator
- Enter your numbers in the Number Sequence Calculator above.
- Press the Generate button to see the result.
- To try other numbers, change any value and press the button again.
The sequence formulas
Arithmetic nth: aₙ = a₁ + (n−1)·d · · · Geometric nth: aₙ = a₁·r^(n−1) · · · Fibonacci: aₙ = aₙ₋₁ + aₙ₋₂
Arithmetic adds a constant difference d each term. Geometric multiplies by a constant ratio r. Fibonacci adds the two terms before. The sum of series is the total of all n terms in the list. The calculator works from the decimals exactly as you type them, takes 1 to 50 terms, and rounds each term and the sum to at most 6 decimal places. A value of 10^15 or more is written in scientific form, such as 1.11111 × 10^49. Fibonacci takes up to 500 terms and prints every digit of a whole number.
Worked example
Arithmetic with a₁=3, d=4 and 10 terms: 3, 7, 11, 15, 19, 23, 27, 31, 35, 39. 10th term: 3 + 9×4 = 39. Sum: 210. Geometric with a₁=2, r=3 and 6 terms: 2, 6, 18, 54, 162, 486. 6th term: 2 × 3⁵ = 486. Sum: 728. Fibonacci from 0 and 1 with 10 terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. 10th term: 21 + 13 = 34. Sum: 88. The last term divided by the one before is 34 ÷ 21 = 1.619048.
Frequently asked questions
How do I find the sum of a sequence?
Arithmetic: S = n(a₁ + aₙ)/2. Geometric (r ≠ 1): S = a₁(1 − rⁿ)/(1 − r). For infinite geometric with |r|<1: S = a₁/(1 − r).
What is the nth term in the result?
It is the last term in the list, the one whose position equals the number of terms you entered. Ask for 10 terms and the nth term is the 10th.
How do I find a Fibonacci number?
Pick Fibonacci, leave the starting terms at 0 and 1, and enter how many terms you want. The nth term is the last one in the list. With 0 and 1 as the start, the 101st term is 354,224,848,179,261,915,075, and every digit is exact. Some books start from 1 and 1, which moves each term one place to the left, so enter 1 and 1 to match them.
Can I change the first two Fibonacci terms?
Yes. Any two starting terms work, decimals and negative numbers too. 2 and 1 gives the Lucas numbers: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76.
Why does the ratio settle near 1.618?
In any Fibonacci-style sequence with positive terms, the last term divided by the one before gets closer and closer to the golden ratio, (1 + √5) ÷ 2 = 1.618034 to 6 decimals. The calculator shows that ratio under the sum. After 20 terms from 0 and 1 it already reads 1.618034.
Can the calculator handle other sequences?
It builds arithmetic, geometric and Fibonacci-style sequences. A sequence with its own rule, such as the factorials or the square numbers, needs a spreadsheet. The big number calculator gives a single factorial.