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Mathematics

Arithmetic and Geometric Sequences

A High School & College Primer on Patterns, Formulas, and Sums

Sequences and series show up on almost every Algebra 2 and Precalculus exam — and most textbooks bury the key ideas under pages of definitions before showing a single worked number. This guide skips the padding.

**TLDR: Arithmetic and Geometric Sequences** covers everything a student needs: what sequences are and how to read the notation, the explicit and recursive formulas for arithmetic and geometric patterns, Gauss's trick for summing a list of numbers, partial sum formulas for finite series, and the convergence rule for infinite geometric series (including how repeating decimals are secretly infinite series in disguise). The final section ties it all together with real modeling problems — savings plans, depreciation, stadium seating — and shows exactly where these ideas connect to functions and calculus.

This primer is written for students in grades 9–12 working through Algebra 2 or Precalculus, early college students filling in gaps before a calculus course, and parents or tutors who need a clean, fast reference. If you have been searching for a clear sequences and series algebra 2 resource that gets to the point, this is it. Every section leads with the one sentence that matters most, then unpacks it with concrete examples and worked numbers. No filler, no fluff.

At 10–20 pages, it is short by design — long enough to build real understanding, short enough to read the night before a test.

Pick it up, work the examples, and walk into your next exam ready.

What you'll learn
  • Identify whether a sequence is arithmetic, geometric, or neither, and find its common difference or ratio.
  • Write explicit and recursive formulas for arithmetic and geometric sequences and use them to find any term.
  • Compute partial sums of arithmetic and geometric sequences using the standard formulas.
  • Determine whether an infinite geometric series converges and find its sum when it does.
  • Translate word problems (savings, depreciation, population, seating) into sequence and series models.
What's inside
  1. 1. Sequences, Terms, and Notation
    Introduces what a sequence is, how terms are indexed, and the difference between explicit and recursive definitions before specializing to arithmetic and geometric types.
  2. 2. Arithmetic Sequences
    Defines arithmetic sequences via a common difference, derives the explicit and recursive formulas, and works examples of finding terms and missing values.
  3. 3. Geometric Sequences
    Defines geometric sequences via a common ratio, derives the explicit formula, and contrasts exponential growth and decay with arithmetic patterns.
  4. 4. Sums: Arithmetic and Finite Geometric Series
    Introduces sigma notation and derives the partial sum formulas for arithmetic and finite geometric series, with worked examples including Gauss's trick.
  5. 5. Infinite Geometric Series and Convergence
    Explains when an infinite geometric series has a finite sum, derives the formula a_1/(1-r) for |r|<1, and shows applications like repeating decimals.
  6. 6. Modeling with Sequences: Where This Shows Up
    Applies arithmetic and geometric models to real situations like savings plans, depreciation, simple population models, and stadium-seating problems, and previews the bridge to functions and calculus.
Published by Solid State Press
Arithmetic and Geometric Sequences cover
TLDR STUDY GUIDES

Arithmetic and Geometric Sequences

A High School & College Primer on Patterns, Formulas, and Sums
Solid State Press

Arithmetic and Geometric Sequences

A High School & College Primer on Patterns, Formulas, and Sums

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Who This Book Is For

If you're sitting in Algebra 2 or Precalculus and sequences and series finally broke your brain, this book is for you. It's also for the student who needs a fast high school math series and sequences review before a chapter test, a final, or the SAT/ACT math section — and for the parent or tutor who wants a clean, reliable reference to work from.

This arithmetic and geometric sequences study guide covers every core idea: term notation, the common difference and common ratio, explicit and recursive formulas, arithmetic series, the geometric series sum formula explained step by step, and infinite geometric series convergence explained with real intuition. Think of it as algebra 2 sequences formulas quick reference — about 15 focused pages, nothing padded.

Read it straight through. Every section builds on the last, so don't skip ahead. Work through each worked example yourself before reading the solution, then use the precalculus sequences practice problems at the end to find out what actually stuck — and what needs another pass — before your exam.

Contents

  1. 1 Sequences, Terms, and Notation
  2. 2 Arithmetic Sequences
  3. 3 Geometric Sequences
  4. 4 Sums: Arithmetic and Finite Geometric Series
  5. 5 Infinite Geometric Series and Convergence
  6. 6 Modeling with Sequences: Where This Shows Up
Chapter 1

Sequences, Terms, and Notation

A sequence is an ordered list of numbers. Order matters: the sequence 1, 3, 5, 7, … is different from 7, 5, 3, 1, … even though both contain the same values. Each number in the list is called a term, and the position it occupies is its index (also called a subscript). The first term sits at index 1, the second at index 2, and so on.

The standard notation writes the term at index $n$ as $a_n$. So $a_1$ is the first term, $a_2$ is the second, $a_{10}$ is the tenth. The whole sequence is sometimes written $\{a_n\}$ or simply listed out: $a_1, a_2, a_3, \ldots$ The index $n$ is almost always a positive integer — think of it as a counter ticking up from 1.

A common mistake is to treat $a_n$ like a function of a continuous variable, the way you might think of $f(x)$. Sequences are discrete: $n$ jumps 1, 2, 3, … and there is no $a_{2.5}$. That said, the parallel to functions is real and useful — you will see it made precise when the course reaches series in calculus.

Two ways to define a sequence

There are two fundamentally different ways to give a rule for a sequence, and both appear constantly.

An explicit formula (also called a closed-form formula) gives $a_n$ directly in terms of $n$. You plug in any index and get that term immediately, with no need to know earlier terms.

A recursive formula (or recurrence relation) defines each term by referring back to one or more previous terms. To use it, you need a starting value — called the initial condition or seed — and then you apply the rule repeatedly to build the sequence forward.

Neither form is universally better. Explicit formulas are faster when you need a specific term far out in the sequence. Recursive formulas often reveal why the sequence behaves the way it does, because they describe the mechanism that generates each next term.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

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