In this article, we look at Sequences and Series – definition, types and Worked 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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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 $n≥1$. $U_n$ 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, $U_n=a +(n-1)d$.

The general term for obtaining the terms in an Arithmetic Progression or linear Sequence is

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

where

$n$=position number of a term and n≥1

$a$=first term

$d$=common difference (between consecutive terms)

The Common Difference (d)

The common difference is obtained by subtracting a preceding term from a consecutive term. For example, given the sequence $2,4,6,8,10,…$

$d=4-2$ or $6-4$ or $8-6$ or $10-8$ etc

Therefore, $d=U_2-U_1$ or $U_3-U_2$ or $U_4-U_3$ … $U_n -U_{n-1}$ or $U_{n+1} – U_n$.

Finding a term in an A.P.

Example 1:

Find the 9th term of the Arithmetic Progression (A.P.) $3,7, 11, 15,…$

Solution

The Sequence is $3,7, 11, 15,…$

First term (a)=3

common difference (d)$= 7-3=4$

position number of term (n) = 9 [Note: 9th term]

The general term (nth term) of an A.P. is given by

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

$U_9= 3 + (9-1)(4)$

$= 3 + (8)(4)$

$= 3 + 32$

$=35$

Example 2:SSSCE July 2002 OBJ Q5

A sequence is given as $-5,-2,1,4,…$. Find the $23rd$ term.

Solution

The Sequence is $-5,-2,1,4,…$

First term (a)$=-5$

common difference (d)$= -2-(-5)=3$

position number of term (n) = 23 [Note: 23rd term]

The general term (nth term) of an A.P. is given by

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

$U_9= -5 + (23-1)(3)$

$= -5 + (22)(3)$

$= -5 + 66$

$=61$

Example 3: SSSCE July 2003 OBJ Q19

Write down the 27th term of the sequence 1951, 1954, 1957, 1960 …

Solution

The Sequence is $1951, 1954, 1957, 1960 … $

First term (a)$=1951$

common difference (d)$= 1954-1951=3$

position number of term (n) = 27

The general term (nth term) of an A.P. is given by

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

$U_9= 1951 + (27-1)(3)$

$= 1951 + (26)(3)$

$= 1951 + 78$

$=2029$

Example 4: WASSCE Nov 2014 OBJ Q8

Find the $26th$ term of the sequence $1909, 1913, 1917, 1921…$.

Solution

The Sequence is $1909, 1913, 1917, 1921… $

First term (a)$=1909$

common difference (d)$= 1913-1909=4$

position number of term (n) = 26

The general term (nth term) of an A.P. is given by

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

$U_9= 1951 + (26-1)(4)$

$= 1909 + (25)(4)$

$= 1909 + 100$

$=2009$

Technology (Using the Scientific Calculator)

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