Sequences — Introduction

Arithmetic sequences, geometric sequences, and the $\Sigma$ notation (high-school level)

Overview

Arithmetic Geometric Efficiently compute sums of sequences
$a_n = a + (n-1)d$
$a_n = a r^{n-1}$
$\Sigma$ notation and formulas
$\sum k,\ \sum k^2,\ \sum k^3$

Figure 1: Relationship among introductory topics — arithmetic and geometric sequences lead into the $\Sigma$ notation.

In this introduction, we cover the basics of sequences as taught in high school: the general term and sum formulas for arithmetic and geometric sequences, together with the computational techniques provided by the $\Sigma$ notation.

Learning goals

  • Understand and prove the general-term and sum formulas for arithmetic sequences.
  • Understand and prove the general-term and sum formulas for geometric sequences.
  • Master the use of the $\Sigma$ notation and its basic formulas ($\displaystyle\sum k$, $\displaystyle\sum k^2$, $\displaystyle\sum k^3$).

Table of contents

  1. Chapter 1 Arithmetic sequences

    General term, sum formula, arithmetic mean.

  2. Chapter 2 Geometric sequences

    General term, sum formula, geometric mean, infinite geometric series.

  3. Chapter 3 $\Sigma$ notation and formulas

    Definition of $\Sigma$ and proofs of $\displaystyle\sum k$, $\displaystyle\sum k^2$, $\displaystyle\sum k^3$.

Prerequisites

  • Algebraic manipulation and factoring.
  • Basic equations and inequalities.

Key formulas

Arithmetic sequence

$$a_n = a + (n-1)d$$ $$S_n = \dfrac{n(a + l)}{2} = \dfrac{n\{2a + (n-1)d\}}{2}$$

Here $a$ is the first term, $d$ is the common difference, and $l = a_n = a + (n-1)d$ is the last term (the $n$-th term).

Geometric sequence

$$a_n = ar^{n-1}$$ $$S_n = \dfrac{a(1 - r^n)}{1 - r} \quad (r \neq 1)$$

Here $a$ is the first term and $r$ is the common ratio (the ratio between consecutive terms).

$\Sigma$ formulas

$$\displaystyle\sum_{k=1}^{n} k = \dfrac{n(n+1)}{2}$$ $$\displaystyle\sum_{k=1}^{n} k^2 = \dfrac{n(n+1)(2n+1)}{6}$$ $$\displaystyle\sum_{k=1}^{n} k^3 = \left\{\dfrac{n(n+1)}{2}\right\}^2$$