Sequence and Series | Algebra | Mathematics

Sequence and Series – Definition, Types, Examples

In this article, we look at Sequences and Series – definition, types and Worked Examples.

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Sequence and Series - Definition, Types, Examples

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Sequence is a set of ordered numbers
Finite sequence ends. Infinite sequence continues to infinity.
Series is when the terms in a sequence are connected by +ve or -ve or both

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Finding the position number of a term and number of terms in Arithmetic Progression

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A sequence is a set of numbers (or terms) arranged in a defined order such that there is a rule for obtaining the numbers. For example, $1,2,3,4,5,...$ is a sequence of numbers. We observe that each succeeding term increases by 1. That’s, we can obtain terms in the sequence by adding 1 to the preceding term. The Sequence can be descending too. For example, $45, 41, 37, 33, 29,...$ is a sequence.

Finite and Infinite Sequence

A finite sequence is a type of sequence whose last term is known. For example, $2,4,6,8,10$ is a finite sequence. The last term is 10.

Infinite Sequence

Infinite Sequence is a type of sequence whose last term is not known. For example, $2,4,6,8,10,...$ is an infinite sequence..The three dots (…) indicate the sequence continues to infinity.

Series

A series is formed when the terms in a sequence are connected by either +ve or -ve or both. For example, $1+2+3+4+5+..., 1+2-3+4-5+6-7...$ are series.

Finite and Infinite Series

A finite series is a type of series whose last term is known. For example, $2+4+6+8+10$ is a finite series. The last term is 10.

Infinite Series

Infinite Series is a type of series whose last term is not known. For example, $2,4,6,8,10,...$ is an infinite series..The three dots (…) indicate the series continues to infinity.

Types of Progression

We will consider five types of progressions. These are Arithmetic Progression (A.P.), Geometric Progression (G.P.), Arithmetico-Geometric progression (AGP), Harmonic Progression (H.P.) and Fibonacci Sequence.

Arithmetic Progression (A.P)

An Arithmetic Progression is a progression where the algebraic difference between any two consecutive terms is constant or the same throughout. For example, $3, 5, 7, 9, ...$ is an Arithmetic Progression (A.P.). The constant difference between any two consecutive terms is 2. This is called the common difference.

An A.P. is also called a linear sequence. This is because the constant difference between consecutive terms in an A.P. creates a straight line when plotted on a graph, hence the term “linear.”

In other words, the relationship between the terms in an AP can be represented by a linear equation of the form $y=mx+c$ , where

$m$ represents the common difference (slope) between consecutive terms, and

$c$ represents the initial term (y-intercept).

Terms in an A.P.

Let $U_n$ represent a term in an A.P. Thus, U1 is first term, $U_2$ is second term, $U_3$ is third term etc.

$U_n$ is any term where $n\ge 1$ . $U_n$ is called the $nth$ term. This means when $n=1$ , we have $U_1$ , first term. When $n=2$ , we have $U_2$ , second term.

The first term in an A.P. , $U_1$ , is normally represented by $a$ . That’s, $U_1=a$ .

$=> U2= a + d$

$U3= a + d + d = a+2d$

$U4 = a + d + d + d=a+3d$

$U5= a + d + d + d + d = a+4d$

$U6= a + d + d + d + d +d= a+5d$

$\vdots$

$Un= a + (n-1)d$

We observe that the first term ( $U_1$ ) has zero $d$ , the second term ( $U_2$ ) has 1 $d$ . The third term ( $U_3$ ) has $2d$ , as in $U_2=a+2d$ ; the fourth term ( $U_4$ ) has $3d$ , as in $a+3d$ . This implies that $U_n$ term will have (n-1) $d$ . That’s, $U_n=a +(n-1)d$ .