Arithmetic Progression

Arithmetic progression, also known as A.P. is a sequence in mathematics where the difference between the two consecutive terms is a constant. The constant is known as the common difference. The arithmetic progression Contains the word arithmetic, which is an elementary branch of mathematics. Arithmetic probably has the longest history during the time. It is a method of calculation that is been in use since 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 basic operations of arithmetic like addition, subtraction, multiplication, and division, arithmetic also includes advanced computing of percentages, logarithm, exponentiation and square roots, etc. Arithmetic is a branch of mathematics concerned with numerals and their traditional operations.

What is Arithmetic Progression?

Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive numbers is a constant value. In other words, arithmetic progression can be defined as “A mathematical sequence in which the difference between two consecutive terms is always a constant“. 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 2 is equal to. In AP, three main terms are Common difference (d), nth Term (a_n), and Sum of the first n terms (S_n); all three terms represent the properties of AP. Let’s take a look at what the common difference is in detail,  

We come across different words like sequence, series, and progression in AP; now, let’s see what each word defines,

• 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 to which the sequence corresponds. 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, etc.
• 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 progression.

Note: There are majorly three types of progression:

1. Arithmetic progression (AP)
2. Geometric progression (GP)
3. Harmonic progression (HP)

Notations in Arithmetic Progression

We will come across the following notations in arithmetic progression:

• First term ⇢ a
• Common difference ⇢ d
• Nth term ⇢ a_n
• Sum of first n terms ⇢ S_n

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 Arithmetic Progression

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 a_1, 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 the common difference is,

d = (a_{n + 1} – a_n) or d = (a_n – a_n-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.

First Term of Arithmetic Progression

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

Where,  

a = first term of AP

Nth term of Arithmetic Progression

The nth term can be found by using the formula mentioned below,

a_n = a + (n − 1)d

Where,  

a = First term of AP

d = Common difference

n = number of terms

a_n = 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 an Arithmetic Progression

The Arithmetic progression sum formula is explained below; consider an AP consisting of “n” terms.

S = n/2 [2a + (n − 1) d]

Sum of Arithmetic progression when the First and Last Term is Given,

S  = n/2 (first term of AP + last term of AP)

Or

S = N/2[a+ a_n]

Summary of Arithmetic Progression

• 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,…
• The general form of Arithmetic progression is a, a + d, a + 2d, a + 3d …
• The formula for nth term of Arithmetic progression is a_n  = a + (n – 1) d.
• The sum of first n terms or the Arithmetic sum formula is S_n = n/2[2a + (n – 1) d], S_n = n/2[a + a_n].

Solved Examples on Arithmetic Progression

Example 1: Find the AP if the first term is 15 and the common difference is 4.

Solution:

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…

Example 2: Find the 20th term for the given AP: 3, 5, 7, 9, …

Solution:  

Given, 3, 5, 7, 9, 11……

Here,

a = 3, d = 5 – 3 = 2, n = 20

a_n = a + (n − 1)d

a₂₀ = 3 + (20− 1)2

a₂₀ = 3 + 38

a₂₀ = 41

here 20th term is a₂₀ = 41

Example 3: Find the sum of the first 20 multiples of 5.

Solution:

The first 20 multiples of 5 are 5, 10, 15, … 100.

Here, it is clear that the sequence formed is an arithmetic sequence where,

a = 5, d = 5, a_n = 100, n = 20.

S_n = n/2 [2a + (n − 1) d]

S_n = 20/2 [2 × 5 + (20 − 1)5]

S_n = 10 [10 + 95]

S_n = 1050.

FAQs on Arithmetic Progression

Question 1: What do you mean by Arithmetic progression?

Answer:

The Arithmetic progression is a sequence of numbers where the two consecutive terms have a common difference. For example: 3, 6, 9, 12, 15,…

Question 2: How do you find the sum of Arithmetic progression?

Answer:

In order to find the arithmetic progression sum, following formulas can be used based on what information is provided:

S  = n/2 (first term of AP + last term of AP)

Or

S = N/2[a+ a_n]

Question 3: What is the difference between Arithmetic progression and Arithmetic series?

Answer:

Arithmetic progression is the number of sequences within any range that provides a common difference. Whereas, the arithmetic series/sequence is the sum of all the terms present in the arithmetic progression.

Question 4: What is the use of Arithmetic progression?

Answer:

The Arithmetic progression is the series that gives a common difference between two consecutive terms. It is used in daily life to generalize a set of patterns. For instance, waiting for a bus, suppose that the buses are moving at a constant speed, with the help of AP, you can tell when will the bus arrive. AP can also be used in making pyramid-like structures, etc.