A sequence of numbers following a special pattern is usually known as a progression. The arithmetic progression is one of the examples of this kind of sequence. In this section, we will learn about arithmetic progression.
Table of Contents
What is an Arithmetic Progression
A sequence of numbers is called an arithmetic progression if the differences between the two consecutive terms are the same. For example, consider the sequence \[1, 8, 15, 22, 29, \cdots.\] Note that the differences of each two consecutive terms are the same which is $7.$ So this sequence is an example of an arithmetic progression.
Remark: The arithmetic progression is abbreviated as AP.
General Form of an Arithmetic Progression
Let $a$ denote the first term of an arithmetic progression. If the difference between consecutive terms is $d$, then we have:
$2$nd term: $a+d$
$3$rd term: $a+d+d$ $=a+2d$
$4$th term: $a+2d+d$ $=a+3d$
and so on. Thus the general form of an arithmetic progression is \[a, a+d, a+2d,a+3d, \cdots .\] In the above general form of an AP, $a$ is called the first term and $d$ is called the common difference.
Terms and Notations
For a general arithmetic progression $a_1, a_2,$ $ a_3, \cdots$, the following notations are used to denote a few useful terms related to this AP:
• $a_1$: first term
• $d$: common difference. So we have $d$ $=a_2-a_1$ $=a_3-a_2=\cdots$
• $a_n$: n-th term
• $S_n$: the sum of the first n terms. So we have $S_n$ $=a_1+$ $a_2+$ $\cdots +a_n$
Examples of Arithmetic Progressions
(i) Lets consider the sequence $7, 7,$ $7, 7, \cdots.$ Here the first term is $a_1=7$ and the difference between the consecutive terms is $d=0$ as $7-7=0.$ So it is an AP with the first term $7$ and with the common difference $0.$ This type of AP is known as a constant AP.
(ii) Next, consider $15, 10,$ $5, 0,$ $-5, \cdots.$ Note that the first term is $a_1=15$ and the common difference is $d=-5$ as $15-10$ $=10-5$ $=0-5$ $\cdots=-5.$ So it is an example of an AP with a negative common difference.
(iii) Similarly, the sequence $0, 5, 10,$ $15, \cdots$ is an AP with $a_1=0$ and $d=5.$
From the above three examples, we can conclude that the common difference of an arithmetic progression can be zero, negative and positive. ♣
Also Read:
| Geometric Progression (GP): Definition, Formula, Sum, N-th term, and Common difference with Solved Examples are discussed here. |
| Surds: We discuss the definition of surds with their orders, properties, types, and a few solved examples. |
| Indices: Click here for the definition, and laws of indices with some solved examples. |
| Logarithm: The definition of logarithm with their rules, and formulas are discussed here with a few solved examples. |
n-th term of an Arithmetic Progression
Let us consider a general arithmetic progression $a_1, a_2,$ $ a_3, \cdots.$ If $d$ is the common difference of this AP, then we have the following relation:
$d=a_2-a_1$ $=a_3-a_2=$ $\cdots=a_n-a_{n-1}$
So we can deduce that
The first term is $a_1$ $=a_1+(1-1)d$
The second term $a_2$ $=a_1+d$ $=a_1+(2-1)d$
The third term $a_3$ $=a_2+d$ $=(a_1+d)+d$ $=a_1+(3-1)d$
The fourth term term $a_4$ $=a_3+d$ $=(a_1+2d)+d$ $=a_1+(4-1)d$
The fifth term term $a_5$ $=a_4+d$ $=(a_1+3d)+d$ $=a_1+(5-1)d$
$\quad \quad \vdots \quad \vdots$
The n-th term $a_n=a_1+(n-1)d$
Hence, the n-th term of an AP with the first term $a_1$ and with the common difference $d$ is given as follows: $a_n=$ $a_1+(n-1)d.$ In other words,
the n-th term of an AP = first term $+ (n-1) \times$ common difference.
Sum of the terms of an Arithmetic Progression
Let an AP has the first term $a$ and the common difference $d.$ Then the sum of the first n terms of the AP is given below:
$S_n=\frac{n}{2}[2a+(n-1)d]$
Proof: Note that the AP has the form $a,$ $a+d,$ $a+2d, \cdots .$ Its n-th term $a_n=$ $a+(n-1)d.$ As $S_n$ denotes the sum of the first $n$ terms, we have:
$S_n=a +$ $(a+d)+$ $(a+2d) +$ $\cdots + [a+(n-2)d]$ $+[a+(n-1)d]$ $\cdots$ (I)
Writing the above sum in the reverse order, we obtain that
$S_n=[a+(n-1)d] +$ $[a+(n-2)d]+$ $\cdots + (a+d)$ $+a$ $\cdots$ (II)
(I) + (II) termwise, we deduce that
$2S_n=[2a+(n-1)d]+\cdots +$ $[2a+(n-1)d]$ (n terms)
$\quad \,\, =n[2a+(n-1)d]$
∴ $S_n=\frac{n}{2}[2a+(n-1)d]$
Remark: For an AP with the first term $a$ and the common difference $d$, the sum of the first n terms is given as follows:
$S_n=\frac{n}{2}[2a+(n-1)d]$
$\quad=\frac{n}{2}[a+a+(n-1)d]$
$\quad=\frac{n}{2}[a+\{a+(n-1)d\}]$
$\quad=\frac{n}{2}[\text{first term}+\text{n-th term}]$
∴ If the n-th term is the last term of the AP, then the sum of the terms of that AP = n/2(first term + last term). ♣
Arithmetic Mean
Let $z$ be the arithmetic mean of $a$ and $b.$ Then $a, z$ and $b$ form an AP. So we must have that the common difference is equal to
$z-a=b-z$
⇒ $2z=a+b$
⇒ $z=\frac{a+b}{2}$
∴ the arithmetic mean of two numbers is half of the sum of the two numbers.
Remark: If $a, b$ and $c$ are in AP, then $c$ is the arithmetic mean of $a$ and $b.$ ♣
Formulas of an Arithmetic Progression
We will list all the arithmetic progression formulas (AP formulas) in one place.
• Let $a$ denote the first term of an arithmetic progression (AP) with a common difference $d.$ Then we have the following:
(i) The AP has the form $\{a, a+d, a+2d, \cdots\}$
(ii) n-th term $=a+(n-1)d$
(iii) Sum of the first n terms $=\frac{n}{2}[2a+(n-1)d].$
(iv) If the last term of the AP is $l$, then the sum of the terms of the AP $= \frac{n}{2}(a+l).$
• If three terms form an AP, then we should assume the numbers as
$a-d,$ $a$ and $a+d$
• If four numbers form an AP, then one can assume the numbers as
$a-3d,$ $a-d,$ $a+d$ and $a+3d$
• The sum of the first n natural numbers $=\frac{n(n+1)}{2}$
• The sum of the squares of the first n natural numbers $=\frac{n(n+1)(2n+1)}{6}$
• The sum of the cubes of the first n natural numbers $=\{\frac{n(n+1)}{2}\}^2$
Solved Problems of Arithmetic Progression
| Problem 1: If an AP has the first term $4$ and the common difference $5$, then write down few terms of the AP. |
Solution:
First term $=4$, Second term $=4+5$ $=9$
Third term $=9+5$ $=14$, Fourth term $=14+5$ $=19$
Fifth term $=19+5$ $=24$, Sixth term $=24+5$ $=29$
and so on. So the AP is $\{4, 9, 14, 19, 24, 29, \cdots\}$ ♣
| Problem 2: Find the tenth term of the AP $\{12, 6, 0, \cdots\}$ |
Solution:
Here the first term $a=12$ and the common difference $d=$ $6-12$ $=-6$
So the tenth term is
$a_{10}$$=a+(10-1)d$ $=a+9d$ $=12+9 \times (-6)$ $=12-54$ $=-42$ ♣
| Problem 3: If the sum of second and tenth terms of an arithmetic progression is equal to $12,$ find the sum of fourth, sixth, and eighth terms. |
Solution:
