Number Sequence Calculator

Related Calculators

Sequence type: The family of pattern you are using, such as arithmetic, geometric, or Fibonacci.

First term (a1): The starting number in the sequence.

Common difference or ratio: In an arithmetic sequence, this is the amount added each step. In a geometric sequence, this is the amount multiplied each step.

Term number (n): The position of a term in the pattern, such as 1st, 2nd, or 10th term.

Nth term value (a_n): The actual value at position n.

Explicit rule: A formula that lets you find a term directly from its position instead of listing every earlier term.

Sum through n: The total of the first n terms.

Pattern check from entered terms: A quick test that tells whether the terms you typed match an arithmetic pattern, a geometric pattern, both, or neither.

Fibonacci sequence: A sequence where each new term is the sum of the two previous terms [1].

Mistake: Picking the wrong Sequence type , such as Arithmetic when the pattern is really multiplying each step.
Fix: Use Entered terms to check (comma-separated) and compare the result with Pattern check from entered terms before trusting Requested result .

Mistake: Typing a decimal, 0, or negative value for Term number (n) .
Fix: Enter Term number (n) as a positive whole number like 1, 2, 3, or 10.

Mistake: Using the wrong sign in Common difference or ratio , especially for decreasing or alternating patterns.
Fix: Keep the minus sign if the pattern goes down by subtraction or flips sign with a negative ratio, then confirm with Preview of first terms .

Mistake: Forgetting to enter Nth term value (a_n) when solving for Common difference , Common ratio , or Number of terms .
Fix: Fill in Nth term value (a_n) with the target term you want the sequence to reach.

Mistake: Expecting an Infinite sum for a geometric sequence when the ratio is too large in size.
Fix: For geometric Infinite sum , the Common difference or ratio must have absolute value less than 1, or the Important notes output will show that the sum does not converge.

Mistake: Using the wrong Fibonacci starting terms policy and getting a term that is off by one place.
Fix: Match the calculator setting to your class rule, then verify the first few values in Preview of first terms .

• Term number (n) must be a positive integer. The calculator rejects 0, negatives, and decimals for n.
• Common difference, Common ratio, and Number of terms modes need Nth term value (a_n). Without it, those results are not defined.
• For arithmetic Common difference, n must be greater than 1 because dividing by n - 1 would fail at n = 1.
• For arithmetic Number of terms, if the difference is 0, only a target equal to the first term is feasible. Other targets return N/A.
• For geometric real-number solving, Common ratio from first and nth term requires a nonzero first term, n greater than 1, and a positive a_n/a_1 value in this calculator.
• For geometric Number of terms, the calculator uses logarithms, so it requires a nonzero first term, a positive ratio not equal to 1, and a positive a_n/a_1 value. If the computed n is not a whole number, the target does not land exactly on a whole-number term index.
• For geometric Infinite sum, the result exists only when the ratio's absolute value is less than 1 [3]. Otherwise the series does not converge.
• For Fibonacci, only sequence generation, nth term, finite sum, preview, and pattern help are meaningful here. Modes like Common difference, Common ratio, and Infinite sum are not applicable.
• The pattern check needs at least 3 entered terms. If a geometric check would divide by zero, it is reported as inconclusive instead of forcing a wrong answer.
• Preview terms to show is capped to keep the page fast and readable, so very long previews are intentionally limited.

How the calculator works

The calculator uses the selected Sequence type and Solve for mode to choose the correct formula. It then computes the requested result, shows a direct rule when available, and builds a short preview of the sequence so you can check whether the pattern looks right.

Arithmetic formulas

a_n = a_1 + (n - 1)d

S_n = n/2(2a_1 + (n - 1)d)

d = (a_n - a_1)/(n - 1)

n = ((a_n - a_1)/d) + 1

An arithmetic sequence changes by the same added amount each step [4]. If the difference is negative, the terms go down. If the difference is 0, every term is the same.

Geometric formulas

a_n = a_1r^(n - 1)

S_n = a_1(1 - r^n)/(1 - r)

r = (a_n/a_1)^(1/(n - 1))

n = 1 + log(a_n/a_1)/log(r)

S_infinity = a_1/(1 - r)

For geometric finite sums, if r = 1, the calculator switches to S_n = na_1 to avoid dividing by zero. For infinite sums, it only gives a real result when |r| < 1 [3].

Fibonacci method

F_n = F_(n-1) + F_(n-2)

Instead of using a direct closed formula, the calculator builds Fibonacci terms one by one from the chosen starting policy. That keeps the result easy to understand for beginners and matches the definition that each term is the sum of the two before it [1].

Pattern check logic

If you enter at least 3 values in Entered terms to check (comma-separated), the calculator compares consecutive differences to test for an arithmetic pattern and consecutive ratios to test for a geometric pattern. A constant sequence such as 5, 5, 5 fits both tests. If a ratio test would require division by zero, the geometric check is reported as inconclusive rather than treated as valid.

Worked mini-example

Suppose you choose Arithmetic, want the Nth term, and enter First term (a1) = 2, Common difference or ratio = 3, and Term number (n) = 10.

a_10 = 2 + (10 - 1)3

a_10 = 2 + 27 = 29

The first terms are 2, 5, 8, 11, 14, ... so the result makes sense. The sum through 10 is also computed from the arithmetic sum formula, giving 155.

Assumptions used

The calculator works with real-number outputs for the solving steps shown here. Because of that, some geometric inverse problems are blocked when they would require non-real roots or invalid logarithms. It also assumes sequence positions are whole-number indexes, so a computed number of terms must land on a positive integer to be an exact match.