# Chapter 5 – Arithmetic Progressions

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

In your everyday life, you see several things that tend to have a particular pattern. Whether it’s the petals of a flower, the leaves on a branch, the days in a week, or steps of a staircase – every repetitive arrangement helps you generalize the features of these things. Such patterns are studied in mathematics as arithmetic progressions (AP).

In addition to helping in the different aspects of our daily life, the concepts and formulas related to arithmetic progressions prove quite useful in estimations and predictions of the business world and technology. You need to learn the fundamentals of an arithmetic sequence to solve the associated questions.

#### What is Arithmetic Progression?

Arithmetic progression also called an arithmetic sequence, is basically a string of numbers that display a common difference between every two consecutive numbers. For example, the sequence 2, 4, 6, 8, 10, 12, 14 forms an AP where the consecutive numbers differ by ‘2’.

Common Difference in Arithmetic Progression

Every arithmetic progression is identified by the constant difference that is exhibited between its consecutive terms. This difference is called ‘common difference’ and indicated by ‘d’ that can be negative, positive, or zero. Accordingly, the terms of a sequence decrease, increase or remain the same in magnitude.

If a given AP is represented as: x1, x2, x3, x4,… xn, then the common difference can be written as:

d = x2 – x1 = x3 – x2 = x4 – x3 ……. = xn – xn – 1; where ‘n’ is the total number of terms of the AP, and x1, x2,.. are the first term, second term,.. and so on, respectively.

#### General Form of Arithmetic Progression

An arithmetic progression can be expressed as:

x, x + d, x + 2d, x + 3d,… x + (n – 1)d

This is a general arithmetic sequence formula that can be used to find the complete arithmetic progression if the first term ‘x’, the common difference ‘d’, and the total number of terms ‘n’ are given.

#### Properties of Arithmetic Progressions

Every arithmetic progression exhibits the following characteristics:

• If the same quantity/value is subtracted from or added to each term of an AP, the resulting terms of the sequence are also in AP, showing the same common difference as the original one.
• If every term of an AP (with common difference ‘d’) is multiplied with or divided by the same non-zero value (say, z), the resulting terms are also in an AP. However, the new sequence’s common difference will be the ‘zd’, in case of multiplication, and ‘d/z’ in division.
• When a few terms are selected at regular intervals of an AP, the selected terms also form an AP.
• If three numbers p, q, r are in AP, then q is the arithmetic mean of the other two numbers and can be written as: q = (p + r)/2

#### What is an Arithmetic Series?

The sum of the terms of a finite arithmetic sequence is called arithmetic series. So, if an arithmetic sequence is x1, x2, x3, x4,… xn, its arithmetic series is given by:

x1 + x2 + x3 + x4 +… + xn

An alternate form of arithmetic series formula is:

x + (x + d) + (x + 2d) + (x + 3d) +… + (x + (n – 1)d)

where ‘x’ is the first term and ‘d’ is the common difference of the arithmetic sequence.

#### Sum of Arithmetic Series

The sum of an arithmetic series is computed by multiplying the number of terms with the average of the first and last term of the sequence. In other words, it is expressed as:

Sn = n(x1 + xn)/2; where ‘n’ is the number of terms, x1 is the first term, and xn is the last term of the series.

Sum of Popular AP Series

The series of natural numbers, even numbers, and odd numbers commonly feature in the exams. The sum of these series can be easily found with the help of the following formulas:

For the first ‘n’ natural numbers (1, 2, 3,…,n), the sum = n(n+1)/2

For the first ‘n’ even numbers (2, 4, 6,…,n), the sum = n(n + 1)

For the first ‘n’ odd numbers (1, 3, 5,…,n), the sum = n2

#### FAQs

1. What is an arithmetic progression in simple words?

Arithmetic progression refers to an ordered sequence of terms, where all the consecutive terms are equidistant from each other.

2. What are the types of arithmetic progression?

There are two main types of arithmetic progressions, namely finite AP and infinite AP. An arithmetic sequence with a finite number of terms forms a finite AP. On the other hand, an infinite AP has an infinite number of terms. Thus, there is no last term in an infinite AP.

3. How is AP calculated?

The general form of any nth term of the arithmetic progression is: a + (n – 1)d; where ‘d’ is the common difference and ‘a’ is the first term of the series.

With the help of this arithmetic progression formula, you can find any term or whole of the AP.

4. What is the formula for sum of arithmetic progression?

The formula for sum of arithmetic sequence with ‘n’ number of terms is:

Sn = n(2a + (n−1)d)/2; where ‘a’ is the first term, ‘d’ is the common difference, and Sn denotes the sum of ‘n’ terms of an AP.

5. How do you find the sum of an infinite AP?

The sum of an infinite AP is infinity due to the unending nature of series. However, the sum is said to be ‘∞’ if the common difference is greater than zero, and ‘-∞’ if the common difference is less than zero.

For a more detailed understanding of arithmetic progression class 10th concepts with examples, checkout the video lectures on MSVgo.

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