What is an Arithmetic Progression?
Arithmetic probably has the longest history during the time. It is a method of calculation that is been in use from ancient times for normal calculations like measurements, labeling, and all sorts of day-to-day calculations to obtain definite values. The term got originated from the Greek word “arithmos” which simply means numbers.
Arithmetic is the elementary branch of mathematics that specifically deals with the study of numbers and properties of traditional operations like addition, subtraction, multiplication, and division.
Besides the traditional operations of addition, subtraction, multiplication, and division arithmetic also include advanced computing of percentage, logarithm, exponentiation and square roots, etc. Arithmetic is a branch of mathematics concerned with numerals and their traditional operations.
Basic Operations of Arithmetic
Arithmetic has four basic operations that are used to perform calculations as per the statement:
Arithmetic Progression, (AP) is a sequence of numbers in which the difference between any two consecutive numbers is a constant value. For example, the series of numbers: 1, 2, 3, 4, 5, 6,… are in Arithmetic Progression, which has a common difference (d) between two successive terms (say 1 and 2) equal to 1 (2 – 1).
A common difference can be seen between two successive terms, even for odd numbers and even numbers, that is 2 is equal to. In AP, three main terms are Common difference (d), nth Term (an), Sum of the first n terms (Sn), All three terms represent the properties of AP. Let’s take a look at what common difference is in detail,
In other words, arithmetic progression can be defined as “A mathematical sequence in which the difference between two consecutive terms is always a constant“.
We come across the different words like sequence, series, and progression in AP, now let us see what does each word define –
Sequence is a finite or infinite list of numbers that follows a certain pattern. For example 0, 1, 2, 3, 4, 5… is the sequence, which is an infinite sequence of whole numbers.
Series is the sum of the elements in which the sequence is corresponding. For example 1 + 2 + 3 + 4 + 5…. is the series of natural numbers. Each number in a sequence or a series is called a term. Here 1 is a term, 2 is a term, 3 is a term …….
Progression is a sequence in which the general term can be expressed using a mathematical formula or the Sequence which uses a mathematical formula that can be defined as the progression.
General form of arithmetic progression is a, a + d, a + 2d ……… a + (n – 1)d
Here are some examples of AP:
- 6, 13, 20, 27, 34,41, . . . .
- 91, 81, 71, 61, 51, 41,. . . .
- π, 2π, 3π, 4π, 5π,6π ,…
- -√3, −2√3, −3√3, −4√3, −5√3, – 6√3,…..
Common Difference of an A.P.
The common difference is denoted by d in arithmetic progression. It’s the difference between the next term and the one before it. For arithmetic progression, it is always constant or the same. In a word, if the common difference is constant in a certain sequence, we can say that this is A.P. If the sequence is a1, a2, a3, a4, and so on.
In other words, the common difference in the arithmetic progression is denoted by d. The difference between the successive term and its preceding term. It is always constant or the same for arithmetic progression. In other words, we can say that, in a given sequence if the common difference is constant or the same then we can say that the given sequence is in Arithmetic Progression (AP).
The formula to find common difference is d = (an + 1 – an) or d = (an – an-1).
If the common difference is positive, then AP increases. For Example 4, 8, 12, 16….. in these series, AP increases
If the common difference is negative then AP decreases. For Example -4, -6, -8……., here AP decreases.
If the common difference is zero then AP will be constant. For Example 1, 2, 3, 4, 5………, here AP is constant.
The sequence of Arithmetic Progression will be like a1, a2, a3, a4,…
common difference (d) = a2 – a1 = d
a3 – a2 = d
a4 – a3 = d and so on.
Other important terms that are used to explain the properties of an Arithmetic progression are,
First Term of AP
The Arithmetic Progression can be written in terms of common difference (d) as:
a, a + d, a + 2d, a + 3d, a + 4d, ………., a + (n – 1)d
a = first term of AP
n-th term of AP
The nth term can be found by using the formula mentioned below:
an = a + (n − 1)d
a = First term of AP
d = Common difference
n = number of terms
an = nth term
Note: The sequence’s behavior is based on the value of a shared difference.
If “d” is positive, the terms will increase to positive infinity.
If “d” is negative, the terms of the members increase to negative infinity
Sum of n terms
The formula for the AP sum is explained below, consider an AP consisting of “n” terms.
S = n/2 [2a + (n − 1) d]
Sum of AP when the First and Last Term is Given
S = n/2 (first term of AP + last term of AP)
In short Arithmetic Progression, (AP) is a sequence of numbers in which the difference between any two consecutive numbers is a constant value. For example, the series of numbers: 1, 2, 3, 4, 5, 6,…
a, a + d, a + 2d, a + 3d ……….
the nth term an = a + (n – 1) d
and sum of first nth terms = Sn = n/2[2a + (n – 1) d]
= n/2[a + an]
Question 1: Find the AP if the first term is 15 and the common difference is 4.
As we know,
a, a + d, a + 2d, a + 3d, a + 4d, …
Here, a = 15 and d = 4
= 15, (15 + 4), (15 + 2 × 4), (15 + 3 × 4), (15 + 4 × 4),
= 15, 19, (15 + 8), (15 + 12), (15 + 16), …
= 15, 19, 23, 27, 31, …and so on.
so the AP is 15, 19, 23, 27, 31………..
Question 2: Find the 20th term for the given AP: 3, 5, 7, 9, …
Given, 3, 5, 7, 9, 11……
a = 3, d = 5 – 3 = 2, n = 20
an = a + (n − 1)d
a20 = 3 + (20− 1)2
a20 = 3 + 38
a20 = 41
here 20th term is a20 = 41