Number Sequence CalculatorFree online math calculator — instant results, no sign-up
Terms and sums of arithmetic and geometric sequences.
About the Number Sequence Calculator
This calculator computes the terms and partial sums of arithmetic and geometric sequences. It generates the sequence list, finds the nth term, and calculates the total sum for a specified number of terms. It is designed for students, educators, and anyone working with recurring series in mathematics.
- Type
- The sequence pattern: Arithmetic (constant difference) or Geometric (constant common ratio).
- First term
- The initial value of the sequence (a), can be positive, negative, or zero.
- Difference / ratio
- The common difference added each step (for arithmetic) or the common ratio multiplied each step (for geometric).
- Number of terms
- The total count of terms (k) to compute; a positive integer.
Formula and method
Arithmetic: a_n = a + (n - 1) * d and S_k = (k / 2) * (2 * a + (k - 1) * d) | Geometric: a_n = a * d^(n - 1) and S_k = a * (1 - d^k) / (1 - d) (or S_k = a * k if d = 1)
- aFirst term of the sequence
- dCommon difference (arithmetic) or common ratio (geometric)
- kTotal number of terms evaluated
- a_nValue of the sequence at term index n
- S_kSum of the first k terms
The tool evaluates each term sequentially up to a maximum display limit of 50 terms. The total sum is computed analytically using standard closed-form summation formulas, handling the special case of a common ratio equal to 1 in geometric series. Final values are rounded to 6 decimal places.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- TypeArithmetic
- First term2
- Difference / ratio3
- Number of terms10
Results
- nth term29
- Sum155
- Terms2, 5, 8, 11, 14, 17, 20, 23, 26, 29
How to use the Number Sequence Calculator
- 1Choose whether the sequence is Arithmetic or Geometric.
- 2Enter the starting value (First term) and the step increment or multiplier (Difference / ratio).
- 3Specify the total count of terms (Number of terms) to analyze.
- 4Read the generated term list, the value of the final displayed term, and the analytical sum.
Reading your result
The 'Terms' field lists each value in progression, up to the first 50 values. The 'Sum' provides the exact total addition of all k terms based on standard summation formulas. The 'nth term' displays the value of the last generated term in the series list.
Limitations
The sequence display list and displayed nth term are capped at the first 50 terms, even if a higher number of terms is entered for the sum. It does not compute infinite series convergence (such as infinite geometric sums where |d| < 1), nor does it validate non-integer term counts.
Common questions
What happens if the common ratio in a geometric series is 1?+
When the ratio is 1, every term equals the first term. The calculator identifies this and computes the sum simply as the first term multiplied by the number of terms.
Can I use negative numbers or decimals for the difference or ratio?+
Yes. Negative values will produce alternating signs in geometric sequences or decreasing values in arithmetic sequences. Decimal increments and ratios are also supported.
Why does the list of terms stop at 50 when I enter a larger number?+
The display list is capped at 50 elements to prevent browser performance issues, but the 'Sum' calculation still reflects the entire count entered.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
Further reading
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Results are estimates for informational purposes and are not financial, medical or legal advice.