Sequence Calculator

What is a sequence?

A sequence is an ordered list of numbers, called terms, that follows a rule. Each term has a position, written n, so the first term is a₁, the second a₂, and the term at position n is aₙ. Order matters: 2, 4, 8 and 8, 4, 2 are different sequences.

Three rules cover most of what people need in practice. In an arithmetic sequence you add the same amount each step. In a geometric sequence you multiply by the same factor each step. In the Fibonacci sequence each term is the sum of the two before it. This calculator handles all three.

Using this calculator

• Sequence type: arithmetic, geometric or Fibonacci.
• Sequence direction: increasing applies the difference or ratio as entered. Decreasing reverses it: an arithmetic sequence subtracts the difference, and a geometric sequence multiplies by the reciprocal of the ratio.
• Starting number: the first term, any real number.
• Common difference: the step that defines the sequence. For a geometric sequence this field holds the common ratio.
• n-th term to obtain: a whole number from 2 up. The first term counts as term 1.

The results show the n-th term, the sum of the first n terms, and a segment of the sequence: the three terms before the n-th, the n-th itself, and the three after it. For the Fibonacci option only the n-th term is shown, because the starting values are fixed at 1 and 1.

Arithmetic sequences

An arithmetic sequence has a constant gap d between neighbouring terms, such as 3, 7, 11, 15 where d = 4. The terms fall on a straight line when plotted against their position.

aₙ = a₁ + (n − 1) × d

Sₙ = n ÷ 2 × (2 × a₁ + (n − 1) × d) = n × (a₁ + aₙ) ÷ 2

Example: start at 3 with d = 4 and ask for the 10th term. a₁₀ = 3 + 9 × 4 = 39, and the sum of the first ten terms is 10 × (3 + 39) ÷ 2 = 210. Choosing the decreasing direction turns d into −4, giving a₁₀ = 3 − 36 = −33 and a sum of −150.

The sum formula is the idea attributed to the young Gauss: pair the first and last terms, the second and second-to-last, and so on. Every pair totals the same amount, which is why 1 + 2 + … + 100 equals 50 pairs of 101, or 5,050.

Geometric sequences

A geometric sequence multiplies by a fixed ratio r at each step, as in 2, 6, 18, 54 where r = 3. Growth of money at a fixed interest rate, population models and repeated halving all follow this pattern.

aₙ = a₁ × r^{n − 1}

Sₙ = a₁ × (1 − rⁿ) ÷ (1 − r), r ≠ 1

Example: start at 2 with r = 3 and take n = 6. a₆ = 2 × 3⁵ = 486, and the sum is 2 × (1 − 3⁶) ÷ (1 − 3) = 728. In decreasing mode the ratio becomes 1 ÷ 3, so the terms shrink: 2, 0.6667, 0.2222 and so on, and the fifth term is about 0.0247 with a running total of about 2.9877.

When r equals 1 every term is identical and the sum formula would divide by zero, so the calculator returns no result for that ratio. If |r| is below 1 the terms shrink towards zero and the total of infinitely many terms settles at a₁ ÷ (1 − r): for a start of 2 and r = 1/3 that limit is 3.

Finding the n-th term of a sequence

If you are handed a few terms and not a rule, first test which pattern fits. Subtract each term from the next: when the differences are all equal, the sequence is arithmetic. Divide each term by the one before: when the quotients are all equal, it is geometric.

Example: for 5, 9, 13, 17 the differences are all 4, so aₙ = 5 + 4 × (n − 1) = 4n + 1. The 20th term is 4 × 20 + 1 = 81 and the first twenty terms add up to 20 ÷ 2 × (5 + 81) = 860. Entering a start of 5, a difference of 4 and n = 20 into the calculator gives the same two numbers.

Remember the exponent in the geometric formula is n − 1, not n: the first term has been multiplied by r zero times. Mixing this up shifts every term by one position.

The Fibonacci sequence

The Fibonacci sequence begins 1, 1 and then adds the previous two terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The tenth term is 55. Some authors start the list with 0, which shifts every index by one, so check the convention when comparing sources.

Fₙ = Fₙ₋₁ + Fₙ₋₂, F₁ = F₂ = 1

The ratio of consecutive terms approaches the golden ratio, about 1.618034. Dividing neighbouring terms shows it quickly: 89 ÷ 55 ≈ 1.6182 and 144 ÷ 89 ≈ 1.6180. A closed form known as Binet's formula expresses Fₙ through that ratio, but for exact whole-number answers adding up the terms is safer.

This tool calculates the Fibonacci number for n up to 70, where the answer is 190,392,490,709,135. Beyond that point results would no longer be reliable in ordinary floating-point arithmetic, so larger values are declined with a message.

Reading the results

The n-th term and the sum are shown in full when they are whole numbers, with thousands separators. Otherwise they are shown with two decimals when that is enough and four decimals when it is not, so 1.5 appears as 1.50 and 0.125 as 0.1250.

The segment lists each term with up to 14 significant digits and no padding. Very large or very small terms switch to exponent form, for example 2.0E+16 for twenty quadrillion, where E+16 means × 10¹⁶. A term that occurs more than once in the segment, as happens when the difference is 0 or the ratio is −1, is listed a single time.

Computers store most decimals approximately, so long chains of decimal steps can differ from exact arithmetic in the last digits. For a precise accounting figure, round the result to the digits that matter.

Related tools

To add a list of numbers that does not follow a pattern use the sum calculator. For growth factors and repeated multiplication see the exponent calculator, and for the typical size of a set of multiplicative factors try the geometric mean calculator. Very large terms can be rewritten with the scientific notation calculator.