We all observe particular patterns in our daily lives. Like, the number of petals in a rose flower, the design of scales on a fish, the shape of a pine cone, etc. If we implement such patterns in mathematics, they are known as progressions. A progression is considered a series or sequence of integers that abide by a particular pattern. There are three types of mathematical progressions: Arithmetic Progression, Geometric Progression, and Harmonic Progression. We shall learn all the important concepts of Arithmetic Progression in this article.
The distance between the steps of a ladder is in arithmetic progression. Arithmetic progression or arithmetic sequence is a set of integers in which the difference between any two numbers is always the same. The difference between two integers in an arithmetic progression is a common difference. It is represented by the lowercase letter ‘d.’ The students must understand that arithmetic progression occurs when the numbers in a list increase or decrease by a constant common difference. Arithmetic progression is a crucial topic as many day-to-day problems are solved using it. Direct questions related to arithmetic progressions are asked in the JEE Main 2022 Exam.
| Table of Contents |
| Identifying Arithmetic Progressions |
| Finding the Common Difference between an AP |
| How To Make an AP? |
| Arithmetic Sequence Formula |
| Sum of Arithmetic Sequence |
| Conclusion |
| FAQs |
Understand the arithmetic progression as a ladder with ten rungs (footholds). The space between two rungs is the same from bottom to top. The area between the first and second rung equals the distance between the second and third rung, and so on. The uniform area in the ladder is the common difference ‘d’.
Identifying Arithmetic Progressions
In many books, arithmetic progression is abbreviated as ‘AP’. ‘AP’ is also the standard acronym for representing arithmetic progressions. In the CBSE 10th Maths, students will often find this abbreviation instead of the entire term. The students must realise that the question is an arithmetic progression and solve it accordingly.
Given below are some series that follows arithmetic progression:
- 0, 3, 6, 9, 12, ……
- 4, 2, 1, 0, -2, …..
- x, x + 5, x + 10, x + 15, …..
Are they all in an arithmetic progression? Below is a detailed series description of which series is in AP and which isn’t.
- The first series 0, 3, 6, 9, 12, …… follows a pattern. That pattern can be determined by using the concept of common differences. First, finding the difference between the first two terms, we get (3 – 0) = 3. Next, finding the difference between the second and third terms, we get (6 – 3) = 3. Going on in the same manner 9 – 6 = 3, 12 – 9 = 3. Hence, there is a constant difference between each term of this series. Thus, 0, 3, 6, 9, 12, …… is an arithmetic progression.
- Following the same procedure for the second series 4, 2, 1, 0, -2, …… The common difference between first and second terms is (2 – 4) = -2. (1 – 2) = -1 is the difference between the second and third terms of the series. Thus there are two different values for the common difference. Hence, this series is not in arithmetic progression.
- The third series is algebraic. The students are often confused about determining the AP of an algebraic series. No matter what variable is in the series, the series will be an AP if the difference is constant.
In the series x, x + 5, x + 10, x + 15
(x + 5) – x = 5
(x + 10) – ( x + 5) = x + 10 – x – 5 = 5
(x + 15) – ( x + 10) = x + 15 – x – 10 = 5
Since the common difference of ‘5’ is prominent in the series, this progression is an AP.
Finding the Common Difference between an AP
The common difference plays a very important role in determining whether a series is an AP or not. It is calculated by subtraction of consecutive terms. If there are two terms ‘x’ and ‘y’ written as (x, y) in a series, then the common difference will be (y – x). For example, the sequence given below is an AP.
k, l, m, n, o, …….
The common difference will either be (l – k), (m – l), (n – m), or (o – n). The second term of the series is subtracted from the first term irrespective of the type of AP. If the series is increasing, then the common difference is positive. Otherwise, if the series is a decreasing series, the common difference is negative. If the common difference is 0, it is not considered an AP.
Example: Find the common difference between the series 10, 7, 4, 1
Ans: The series 10, 7, 4, and 1 is a decreasing series. Hence, the common difference will be negative.
d = (7 – 10) = -3
or
d = (4 – 7) = -3
Since the common difference is constant, the series is an AP.
The converse of the statement mentioned above is also true. If the common difference between the terms of an AP is a positive integer, then the series will be increasing. Otherwise, if the common distinction between the terms of an AP is a negative integer, the series will be a decreasing series.
Remember to always subtract consecutive numbers. The common difference is not attained if a student removes the third term of the series from the first term. Common differences will be found when any term is subtracted with its consecutive succeeding term. Don’t subtract any term with its preceding term.
Looking at the pattern the common difference is formulated as:
AP → n a-1 , n a , n a+1 , n a + 2
d = ( n a – n a-1), where ‘n’ refers to the term and ‘a’ = 2, 3, 4, 5, ……
How To Make an AP?
The students must follow the below-mentioned steps to make an AP.
Step 1: Choose an integer that will serve as the AP’s first term. This will be the starting number.
Step 2: To get the second term, fix a common difference and start adding or subtracting from the first term.
Step 3: To acquire the third term, add or subtract the common difference from the second term.
Step 4: Steps 1-3 must be repeated to get to the nth term and obtain the necessary AP.
Many times questions related to the formation of AP have appeared in JEE Main Mock Tests 2022 and JEE Advanced 2022 Exam. The students must learn the steps mentioned above by heart.
Arithmetic Sequence Formula
All the rudimentary concepts of AP have been discussed so far. The students must now be confident in the basics of AP. The maths concepts and formulas learnt above will help the students understand the arithmetic sequence formula, which is given as:
For increasing series:
a, a + d, a + 2d, a + 3d, ……………., a + (k – 1)d.
Here ‘a’ is known as the first term of the AP, and ‘d’, as discussed above, is the common difference between consecutive terms. (k – 1) is the final term of the AP, and k = {set of whole numbers}
For decreasing series:
a, a – d, a – 2d, a – 3d, ……………., a – (k – 1)d. The terminology remains the same as written above.
If a student is provided with the initial term and the common difference ‘d’ of an AP, then they can evaluate the value of any term of that AP. The below-mentioned formula is called the explicit formula for arithmetic progression and is formulated as
L = a + (k – 1)d
Sum of Arithmetic Sequence
The sum of an arithmetic progression is the last but very important topic. Questions related to the sum of AP have appeared a lot of times in CBSE 10th Maths papers. There are two dominant formulas for determining the sum of series having ‘n’ number of terms. They are given as
Sn = n/2 [ 2a + (n – 1) d ]
And
Sn = n/2 [ a (1 + n) ]
Here, Sn refers to the sum of the series having ‘n’ terms, ‘a’ is the first term of the series, and ‘d’ is a common difference.
There are two approaches to finding the sum of an arithmetic sequence. If a student knows the first and last terms of the AP, they can use the second formula to determine the sum directly. But, if the students are provided with the nth term, they have to use the first method to obtain the sum.
Example: Find the sum of arithmetic sequence -1, 1, 3, 5, … up to the first 22 terms.
Solution: Given
First term a = -1
n = 22
d = (1 – (-1)) = 1 + 1 = 2
Using the first formula: S22 = n/2 [ 2a + (n – 1) d ]
S22 = 22/2 [ (2 x -1) + (22 – 1) x 2 ]
S22 = 11 x [ (-2) + (21) x 2 ]
S22 = 11 x [ (-2) + 42 ]
S22 = 11 x [ 40 ]
S22 = 440
Answer: The sum of the -1, 1, 3, 5, … up to the first 22 terms is 850.
Conclusion
Arithmetic progression is crucial for the class 10 CBSE board examination. It also serves a decent role in competitive exams like JEE 2022 and other olympiads. The only thing a student needs to remember about arithmetic progression is that it is a series whose pattern is based on the addition or subtraction of common differences. Continuous addition or subtraction of common difference ‘d’ in the terms will lead to the formation of arithmetic series. The students can extend an AP to infinity. That concludes all the concepts related to arithmetic progression. After studying this article, a student can easily solve numerical and word problems present in NCERT Solutions.
FAQs
1. Is it required to learn all of the concepts covered in Chapter 5 of the NCERT Solutions for Class 10 Maths?
Yes, learning all of the subjects in NCERT Solutions for Class 10 for arithmetic progression is required to achieve excellent results in the second term exams. They also concentrate on deciphering Maths solutions in a simple style for kids to comprehend. You can also check the NCERT Solutions for Class 10 Maths for complete solutions to tough and tricky chapters.
2. State the two types of arithmetic progressions?
Generally, there are only two types of arithmetic progressions. They are known as infinite arithmetic progression and infinite arithmetic progression. The finite AP has a starting and an ending term. For example, 1, 6, 11, 16, 21, 26. The infinite AP has a starting term but doesn’t have an ending term. For example, 1, 6, 11, ………….. To show that a series is infinite, we use multiple dots. The ascending and descending APs are derivatives of finite and infinite arithmetic progression.
3. Where can I find revision notes for arithmetic progression?
A student can find relevant notes on Byju’s website. Byju has the most comprehensive collection of study materials available to students of any class. They deliver high-quality information handpicked by qualified instructors with years of classroom experience throughout the country. Students can count on Byju’s for Class 10 Maths as well. Revision Notes for Class 10 Maths for Arithmetic Progressions are available for free download. Students can also access Lakhmir Singh Solutions and many more on the website.
4. What are the major differences between arithmetic progression and geometric progression?
An arithmetic progression (AP) is a sequence of numbers that are produced by the addition of some common constant. This common constant is termed the common difference. It is obtained by subtracting any two consecutive terms. When added, common difference creates an ascending arithmetic progression. The students can also subtract the constant from terms to make a descending series. A series will be called an arithmetic progression if the common difference between the terms is the same. In contrast, a geometric progression (GP) is a numerical sequence in which each successive number is obtained by multiplying a common ratio number.
5. What questions can be formed from arithmetic progressions in the CBSE 10 Board?
A plethora of questions can be formed using the concepts studied in arithmetic progressions. These include multiple-choice questions, descriptive type questions, long answer type questions, short answer type questions, fill in the gaps, and daily life examples. In class 10, CBSE boards, JEE Main Previous Year Question Papers, and JEE Advanced Previous Year Question Papers word problems related to APs are asked multiple times. Students will be able to improve their problem-solving and time management skills by the end of this chapter. This helps the pupils get good grades in their CBSE Term II exams.
6. Is NCERT sufficient for studying arithmetic progressions?
No! Although it is a fact that class 10 NCERT books provide a lump sum of knowledge on arithmetic progression. The books provided by NCERT are written in simple language and cover all the necessary information. But NCERT is not sufficient for dealing with questions that appear in the Board examinations. To be perfect in this chapter, a student must solve reference books. If stuck in any problem, the students can refer to RD Sharma Solutions or RS Aggarwal Solutions.
