Theory

Sequences, progressions and series

How quantities that change at regular intervals are described by sequences, how the terms and sums of arithmetic and geometric progressions are computed, what it means for a sequence to have a limit, and when a sum of infinitely many terms gives a finite number.

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Sequences
Arithmetic progressions

Sequences

1. Sequence

A sequence of real numbers is a function $a : \mathbb{N} \to \mathbb{R}$ that assigns to each natural number $n$ the number $a_n$ , called the term in position $n$ . One writes $(a_n)$ or $a_1, a_2, a_3, \dots$ ; the index can start from $0$ or from $1$ depending on the problem. A sequence is given in two ways:

by its general term, a formula that gives $a_n$ as a function of $n$ , such as $a_n = \dfrac{2n + 1}{n + 3}$ , whose first terms are $\dfrac{3}{4}$ , $1$ , $\dfrac{7}{6}$ , $\dfrac{9}{7}$ ;
by recursion, giving the first term and the rule that obtains each term from the previous one, such as $a_1 = 2$ and $a_{n+1} = 3\,a_n - 1$ , whose first terms are $2$ , $5$ , $14$ , $41$ .

The recursive formula gives the step from one term to the next; the general term makes it possible to compute a distant term without going through all the previous ones.

2. Monotonic and bounded sequences

A sequence is increasing if $a_{n+1} > a_n$ for every $n$ , and decreasing if $a_{n+1} < a_n$ for every $n$ ; with non-strict inequalities it is called non-decreasing and non-increasing, and in all these cases monotonic. To determine this, one studies the sign of the difference $a_{n+1} - a_n$ , or, for positive terms, compares the ratio $\dfrac{a_{n+1}}{a_n}$ with $1$ . The sequence is bounded if there exists a number $M$ such that $|a_n| \leq M$ for every $n$ . For $a_n = \dfrac{2n + 1}{n + 3}$ we have

a_{n+1} - a_n = \frac{2n + 3}{n + 4} - \frac{2n + 1}{n + 3} = \frac{5}{(n + 3)(n + 4)} > 0 ,

so the sequence is increasing; moreover $0 < a_n < 2$ for every $n$ , because $2n + 1 < 2n + 6$ , therefore it is bounded.

Arithmetic progressions

1. Arithmetic progression

An arithmetic progression is a sequence in which each term is obtained from the previous one by adding the same number $d$ , called the common difference: $a_{n+1} = a_n + d$ . Starting from $a_1$ , one adds $d$ once to reach $a_2$ , twice for $a_3$ , and in general $n - 1$ times for $a_n$ , so

a_n = a_1 + (n - 1)\,d .

The progression is increasing if $d > 0$ and decreasing if $d < 0$ , and its terms, plotted as a function of $n$ , lie on a line of slope $d$ . For example, with $a_1 = 3$ and $d = 4$ one obtains the terms $3$ , $7$ , $11$ , $15$ , $\dots$ and the general term is $a_n = 3 + 4\,(n - 1) = 4n - 1$ .

The terms of the arithmetic progression

a_{n} = 4 n - 1

lie on the line

y = 4 x - 1

, whose slope equals the common difference

d = 4

2. Sum of the terms of an arithmetic progression

The sum of the first $n$ terms of an arithmetic progression is

S_n = a_1 + a_2 + \dots + a_n = \frac{n\,(a_1 + a_n)}{2} ,

that is, the number of terms times the mean of the first and the last. For example, the first $100$ natural numbers form an arithmetic progression with common difference $1$ , with $a_1 = 1$ and $a_{100} = 100$ , so

S_{100} = 1 + 2 + \dots + 100 = \frac{100\,(1 + 100)}{2} = 5050 .

Proof

The idea is to add the progression to itself written in reverse order. We write

\begin{aligned} S_n &= a_1 + a_2 + \dots + a_{n-1} + a_n , \\[4pt] S_n &= a_n + a_{n-1} + \dots + a_2 + a_1 . \end{aligned}

In every column the sum is the same, because moving from one column to the next the top term increases by $d$ and the bottom one decreases by $d$ , so every column equals $a_1 + a_n$ . There are $n$ columns, therefore adding the two rows gives $2 S_n = n\,(a_1 + a_n)$ , and dividing by $2$ gives the claim.

The theory continues with 5 more sections (geometric progressions, limit of a sequence, numerical series…), plus all worked examples, the solutions and the PDF.

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