Arithmetic Sequence: nth Term and Sum

What the Arithmetic Sequence Calculator Does

An arithmetic sequence is a list of numbers where each term increases or decreases by the same fixed amount, called the common difference (d). For example, 3, 7, 11, 15 is an arithmetic sequence with d = 4.

This calculator finds two things for you: the value of any term in the sequence (the nth term) and the sum of the first n terms. You enter the first term, the common difference, and the position you care about, and it returns both results.

It is useful for students checking algebra homework, teachers building examples, and anyone working with evenly spaced values such as seating rows, depreciation schedules, or scheduled payments that change by a fixed step.

How It Works: The Arithmetic Sequence Formulas

Two standard formulas drive the results. The nth term tells you the value at any position, and the sum (also called the arithmetic series) adds up the terms from the first through the nth.

• nth term: aₙ = a₁ + (n − 1)d
• Sum of n terms: Sₙ = n/2 × (2a₁ + (n − 1)d)
• a₁ = the first term, d = common difference, n = number of terms
• An equivalent sum formula is Sₙ = n/2 × (a₁ + aₙ), which simply averages the first and last term, then multiplies by how many terms there are.

Worked Example With Real Numbers

Suppose a sequence starts at a₁ = 5 with a common difference d = 3, and you want the 10th term and the sum of the first 10 terms.

nth term: a₁₀ = 5 + (10 − 1) × 3 = 5 + 27 = 32. So the 10th term is 32.

Sum: S₁₀ = 10/2 × (2 × 5 + (10 − 1) × 3) = 5 × (10 + 27) = 5 × 37 = 185. As a check, the averaging form gives 10/2 × (5 + 32) = 5 × 37 = 185, which matches.

Tips and Common Mistakes

Most errors come from small slips in setup, not the arithmetic itself. Watch for these:

• Use (n − 1), not n. The first term already counts as position 1, so only (n − 1) steps are added.
• Find d correctly: subtract any term from the next one (term2 − term1). It must be the same gap throughout, or the sequence is not arithmetic.
• A negative d means the sequence decreases. For example, a₁ = 20, d = −4 gives 20, 16, 12, 8.
• d can be a fraction or decimal (like 0.5 or 1/4). Keep it exact rather than rounding early.
• Make sure n is a positive whole number; you cannot have term 2.5 or term 0.

Factors That Affect the Result

Only three inputs change the output: the first term, the common difference, and the number of terms. The first term shifts the entire sequence up or down. The common difference controls how fast it grows; a larger d makes both the nth term and the sum increase more quickly.

The number of terms has the strongest effect on the sum, because Sₙ grows roughly with n squared. Doubling n more than doubles the total, since you are adding more terms and each added term is larger than the last.