Arithmetic & Geometric Sequences: Patterns With Formulas

A sequence is simply an ordered list of numbers that follows a rule. The two rules the tests care about are the simplest ones: keep adding the same number, or keep multiplying by the same number. Recognize which is happening, identify that fixed number, and a short formula lets you leap to the 50th term without ever writing the first 49.

This guide builds both sequence types from a clear diagram, gives you the nth-term formulas, and finishes with worked and practice problems matched to real test difficulty at Northside Tutoring.

Why Sequences Matter

Sequences test pattern recognition and formula application at once. They also quietly preview bigger ideas: arithmetic sequences are linear functions in disguise, and geometric sequences are exponential ones. The tests reward students who can identify the pattern type fast and apply the right formula instead of listing terms by hand.

What a Sequence Is

Each number in a sequence is a term, labeled by position: a₁ is the first term, a₂ the second, aₙ the nth. The whole game is finding the rule that gets you from one term to the next.

Arithmetic Sequences

An arithmetic sequence increases (or decreases) by the same amount every step — the common difference d. In 3, 7, 11, 15…, d = 4. To find d, subtract any term from the next one.

Geometric Sequences

A geometric sequence multiplies by the same factor every step — the common ratio r. In 2, 6, 18, 54…, r = 3. To find r, divide any term by the one before it.

Telling Them Apart

Check the gaps. If consecutive terms differ by a constant amount, it's arithmetic. If they differ by a constant multiple, it's geometric. The sequence 5, 10, 20, 40 multiplies (geometric, r = 2); 5, 10, 15, 20 adds (arithmetic, d = 5).

Jumping to the nth Term

You rarely want to list out 30 terms. The formulas take you straight there:

The (n − 1) matters: you take d (or r) one fewer time than the term number, because the first term takes zero steps.

Where You'll See This — Test by Test

Sequence formulas are not provided — you must know them. The SAT frames sequences as functions and patterns; the ACT tests the nth-term formulas directly; the SSAT Upper Level asks for the next term or a specified term.

Watch the Lesson

Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.

Worked Example Problems

These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.

Common Mistakes to Avoid

Practice Problems — You Try

Three problems below. Work each before checking the solution.

The Northside Method — How We Teach This 1-on-1

Reading a blog is a great starting point. But there's a meaningful gap between understanding a concept and reflexively applying it under timed conditions. That gap is exactly what our tutors close.

Every Northside student works through a four-step framework:

1. Assessment. We diagnose which specific skills are slowing your student down — not just whether they "get it" in the abstract.
2. Perfect-match coach. We pair them with an elite tutor (we accept only the top 1% of applicants) whose teaching style fits how your student actually learns.
3. Bespoke plan. A roadmap built around your student's target score, target timeline, and current pacing data.
4. Data-driven adjustment. Every session ends with a check on whether the student's accuracy and speed are moving in the right direction.

And if a student meets all eligibility requirements but doesn't hit the defined score improvement? We provide 5 additional hours of cohort learning at no cost. That's the Northside guarantee — built on 25 years of measured outcomes.