Arithmetic sequence: nth term versus sum of the first n terms
Distinguish a single term from a running total, with a step-by-step 5, 8, 11 sequence example.
How it works
An arithmetic sequence changes by the same difference each step. Its nth term is aₙ = a₁ + (n − 1)d. The sum of the first n terms is Sₙ = n(a₁ + aₙ) ÷ 2. The indexing matters: a₁ is the first term, and there are n − 1 jumps to the nth term. The sum formula assumes every term from first to nth is included.
Use the inputs from your own situation and keep the units consistent. A calculator gives an arithmetic estimate under the assumptions you enter; compare it with real records before making a decision.
Worked example
For 5, 8, 11, …, the difference is 3. The tenth term is 5 + 9 × 3 = 32. The first ten terms total 10 × (5 + 32) ÷ 2 = 185. Checking by pairing the first and last terms gives 5 + 32 = 37; five pairs make 185. The answer 32 is a term, not the cumulative sum.
What to check
If the differences vary, this arithmetic formula is inappropriate. A starting index of zero changes the nth-term expression, so translate the problem before entering a₁. For money schedules, confirm whether an initial payment occurs at the start or end and whether interest also applies.
Practical next step
Write the first three terms, calculate successive differences and confirm they match. Decide whether the question asks for one position or all positions through n.
Related calculators and articles
arithmetic sequence, sum series. percentage change vs percentage points.
Save your assumptions and repeat the calculation if an input changes. The linked article explains a neighboring decision that this single formula does not settle.