DIY Cost Tools
Arithmetic Sequence Calculator
Calculate the nth term and sum of an arithmetic sequence from its first term and common difference. Use the prefilled example or enter your own values for immediate results.
First term and common difference may be negative or zero. n must be a whole number from 1 to 1,000,000. Numeric magnitude is limited to 1 trillion.
How to use this calculator
Enter a number in every field, then select Calculate. The default example runs when the page opens. Invalid entries show an explanation and clear previous results. Outputs use up to ten significant digits, so long answers may be rounded.
Formula and explanation
aₙ = a₁ + (n − 1)d. Sₙ = n(a₁ + aₙ)/2. Here a₁ is the first term, d is the constant difference, and n is a positive integer. The sum pairs low and high terms whose averages are identical.
Worked example
Starting at 3 with difference 2 gives 3, 5, 7, …, 21 over ten terms. The sum is 10 × (3 + 21)/2 = 120.
Limitations
Only applies when the difference between consecutive terms stays constant. It does not model geometric sequences, compound growth, or irregular data. Very large values and many terms can lose low-order numerical precision; displayed values are rounded rather than exact symbolic answers. This tool supports learning and planning; verify precision requirements for professional applications.
Frequently asked questions
Can the sequence decrease?
Yes. Enter a negative common difference.
What if the difference is zero?
Every term equals the first term, and the sum is n times that value.
Can n be a decimal?
No. This tool counts discrete terms and requires a positive integer.
Does n = 1 work?
Yes. Both the nth term and sum equal the first term.
Review notes
Reviewed October 1, 2026. Default values and boundary examples were independently checked against the formulas. Blank, nonfinite, excessive, and out-of-range inputs were tested before publication.