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Math · Pre-Calculus

Arithmetic sequences

Find a term, the common difference, or the sum of a sequence that changes by a constant amount each step — every formula here builds off just two numbers: the first term and the common difference. Every question comes with a written walkthrough of which formula applied.

Practise arithmetic sequences nowSee sample questions
3
Difficulty tiers
35s
Target pace
57%
Average first-attempt score

What's covered

  • The nth term formula — apply aₙ = a₁ + (n−1)d, where a₁ is the first term and d is the common difference between consecutive terms.
  • Finding the common difference — subtract any term from the one right after it — the result should be the same no matter which consecutive pair is used.
  • The sum of a finite sequence — apply Sₙ = n/2(a₁ + aₙ), pairing the first and last term of the sequence.
  • Working backward from two given terms — set up a system using the nth term formula for each given term to solve for a₁ and d when neither is given directly.
  • Recognizing an arithmetic sequence — check that the difference between every pair of consecutive terms is constant before applying any arithmetic-sequence formula.

Where students lose marks

Off-by-one errors in (n−1)

The first term uses n=1, which makes (n−1)=0 — plugging n directly into the formula without subtracting 1 shifts every term by one position.

Using the wrong term as a₁ when terms aren't given from the start

If a sequence is described starting from its 5th term, that term isn't a₁ — the nth term formula still needs the actual first term, or has to be adjusted to count from a different index.

Averaging the wrong pair of terms in the sum formula

Sₙ = n/2(a₁+aₙ) needs the first and last term specifically, not the first and second, or any other pair — using the wrong pair changes the average being scaled by n.

Three sample questions

Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.

Sample 01Foundation

Find the 10th term of the arithmetic sequence 4, 9, 14, 19, ...

Sample 02Core

In an arithmetic sequence, the 3rd term is 11 and the 7th term is 27. Find the first term.

Sample 03Advanced

The sum of the first 20 terms of an arithmetic sequence is 670. If the first term is 5, find the common difference.

How to practise this

Arithmetic sequence problems reward tracking exactly which term index you're on — one off-by-one slip in (n−1) shifts every downstream calculation.

Write out a₁, d, and n separately before plugging into a formula

Label the three pieces you have before choosing a formula, and double-check whether a given term is really a₁ or a later term. Untimed practice is where that labeling habit sticks.

Then 35 seconds a question

Move to timed sessions once term-index tracking is automatic. Problems that require solving a system for a₁ and d run longer than direct term lookups.

Fold into a mock

Pair arithmetic sequences with geometric sequences in a mock — the setup logic is nearly identical, just addition versus multiplication.

Stop losing track of n.
Start labeling every term.

  • A formula walkthrough on every item
  • Timed or untimed sessions, any length you like
  • Tracked separately — see your arithmetic sequences accuracy and pace over time
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Related Math subtopics

Geometric sequencesAll of Math