Arithmetic Progressions

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Class 10 Math Chapter 5
Arithmetic Progressions
Important Questions

Here are some important questions for Class 10 Mathematics Chapter 5, Arithmetic Progressions, carefully selected to help students prepare for the CBSE Class 10 Mathematics Examination in 2024-25. These questions cover various types of problems and are designed to assist students in understanding Arithmetic Progressions better. By practicing these diverse question types, students can clarify any doubts they may have and improve their problem-solving skills, leading to better performance in the chapter on Arithmetic Progressions.

Table of Contents

Introduction

In Chapter 5 of Class 10 Mathematics, Arithmetic Progressions, you will study the motivation for studying Arithmetic Progression, the derivation of the nth term and sum of the first n terms of an A.P., and there are applications in solving real-life problems.

What is an Arithmetic Progression (A.P.)?

An Arithmetic Progression is a sequence of numbers in which the difference between consecutive terms remains constant. This constant difference is known as the common difference (d).

Class 10 Arithmetic Progressions Important Questions

Q 1. The sum of first 20 odd natural numbers is :

Options

(a) 100
(b) 210
(c) 400
(d) 420

Ans. (c)

Explanation:
The given A.P. is
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39
Thus, a = 1, d = 2 and n = 20
We know that,
$S_n = \dfrac{n}{a}[2a + (n - 1)d ] \\[4.5 bp] = \dfrac{20}{2}[2(1) + (20 - 1)2 ] \\[4.5 bp] = 10[2+38] = 400$

Q 2. The number of multiples of 4 that lie between 10 and 250 is

Options

(a) 62
(b) 60
(c) 59
(d) 55

Ans. (b)

Explanation:
Multiples of 4 lying between 10 and 250 are 12 , 16, 20, 24……, 248
Here, a = 12, d = 16 - 12 = 4, $a_n$ = 248
          $a_n$ = a + (n - 1)d
⇒     248 = 12 + (n - 1)× 4
⇒     248 - 12 = 4(n - 1)
⇒      $\frac{236}{4}$ = n – 1
⇒      n - 1 = 59
∴      n = 59 + 1 = 60

Q 3. If the sum of n terms of an A.P. is $S_n = 3n^2 + 5n,$ then write its common difference.

Explanation:
We have,
$S_n = 3n^2 +5n \\[4.5 bp] ⇒\space\space\space\space S_{n - 1} = 3(n - 1)^2 + 5(n - 1) \\[4.5 bp] =\space\space\space\space 3(n^2 + 1 - 2n) + 5n - 5 \\[4.5 bp] =\space\space\space\space 3n^2 + 3 - 6n + 5n - 5 \\[4.5 bp] =\space\space\space\space 3n^2 - n - 2 \\[4.5 bp] \text{General term of A.P. is given as,} \\ [4.5 bp] \space\space\space\space T_n = S_n - S_{n-1} \\[4.5 bp] \space\space\space\space= 3n^2 + 5n - 3n^2+ n + 2 \\[4.5 bp] \space\space\space\space= 6n + 2 \\[4.5 bp] \text{ Putting n = 1,2,3, we get }\\ [4.5 bp] \space\space\space\space T_1 = 6 × 1 + 2 = 8 \\[4.5 bp] \space\space\space\space T_2= 6 × 2 + 2 = 14 \\[4.5 bp] \space\space\space\space T_3 = 6 × 3 + 2 = 20 \\[4.5 bp] \text{Now,} \space\space\space\space\space T_2 - T_1 = 14 - 8 = 6 \\[4.5 bp] \text{and} \space\space\space\space\space T_3 - T_2 = 20 - 14 = 6$

∴ common difference = 6

Q 4. Which term of the progression

$20, 19 \dfrac{1}{4}, 18\dfrac{1}{2}, 17\dfrac{3}{4},$ ….. is the first negative term?

Explanation:
Given, A.P. is 20, $19\dfrac{1}{4}, 18\dfrac{1}{2}, 17\frac{3}{4},….. \\[4.5 bp] = 20, \dfrac{77}{4}, \dfrac{37}{2}, \dfrac{71}{4}, \\[4.5 bp] \text{Here,} \space a = 20, d = \dfrac{77}{4} – 20 = \dfrac{77 – 80}{4} = \dfrac{–3}{4} \\[4.5 bp] \text{Let} \space a_n \space \text{ be its first negative term}\\[4.5 bp] \space\space\space\space\space a_n\space < 0 \\[4.5 bp] \text{Then} \space\space\space\space \space a_n + (n – 1)d < 0. \\[5.5 bp] \Longrightarrow \space\space\space\space\space 20+(n – 1)\left(–\dfrac{3}{4}\right) < 0 \\[5.5 bp] \Longrightarrow \space\space\space\space\space 20 \space – \space \dfrac{3}{4}n + \dfrac{3}{4} < 0 \\[5.5 bp] \Longrightarrow\space\space\space\space\space 20 + \dfrac{3}{4} < \dfrac{3}{4}n \\[5.5 bp] \Longrightarrow\space\space\space\space\space \dfrac{83}{4} < \dfrac{3}{4}n \\[5.5 bp] \Longrightarrow\space\space\space\space\space n > \dfrac{83}{4} × \dfrac{4}{3} \\[5.5 bp] \Longrightarrow\space\space\space\space\space n > \dfrac{83}{3} = 27.66...$

28th term will be the first negative term of given A.P.

Q 5. The houses in a row are numbered consecutively from 1 to 49. Show that there exists a value of X such that the sum of numbers of houses proceeding the house numbered X is equal to sum of the numbers of houses following X.

Explanation:
Given, the houses in a row numbered consecutively from 1 to 49.
Now, sum of numbers preceding the number X

= \dfrac{X(X – 1)}{2}

and sum of numbers following the number X

\space\space\space\space\space=\dfrac{49(50)}{2} \space – \space \frac{X(X – 1)}{2} \space – \space X \\[4.5 bp] \space\space\space\space\space= \dfrac{2450 \space – \space X^2 + X \space – \space 2X}{2} \\[4.5 bp] \space\space\space\space\space= \dfrac{2450 \space – \space X^2 \space – \space X}{2} \\[4.5 bp] \text{According to the question,}\\[4.5 bp]

Sum of no’s preceding X = Sum of no’s following X

\dfrac{X(X \space – \space 1)}{2} = \dfrac{2450\space – \space X^2 \space – \space X}{2} \\[4.5 bp] \Longrightarrow \space\space\space\space\space X^2 \space – \space X = 2450\space –\space X^2\space –\space X \\[4.5 bp] \Longrightarrow\space\space\space\space\space 2X^2 = 2450 \\[4.5 bp] \Longrightarrow\space\space\space\space\space X^2 = 1225 \\[4.5 bp] \Longrightarrow\space\space\space\space\space X = 35

Hence, at X = 35, sum of number of houses preceding the house no. X is equal to the sum of the number of houses following X.

CBSE Class 10 Maths Chapter wise Important Questions

Chapter No.	Chapter Name
Chapter 1	Real Numbers
Chapter 2	Polynomials
Chapter 3	Pair of Linear Equations in Two Variable
Chapter 4	Quadratic Equations
Chapter 5	Arithmetic Progressions
Chapter 6	Triangles
Chapter 7	Coordinate Geometry
Chapter 8	Introduction to Trigonometry
Chapter 9	Some Applications of Trigonometry
Chapter 10	Circles
Chapter 11	Areas Related to Circle
Chapter 12	Surface Areas and Volumes
Chapter 13	Statistics
Chapter 14	Probability

Conclusion

If you want to improve your understanding of the concepts in this chapter, you can visit oswal.io. They have a wide range of questions that will help you practice and reinforce what you’ve learned. By solving these questions, you can strengthen your knowledge and become better at solving problems.

Frequently Asked Questions

Q1. How many solutions can a quadratic equation have?

Ans: To identify an Arithmetic Progression, check if the difference between consecutive terms is constant. If the difference between any two consecutive terms is the same throughout the sequence, then it is an A.P.

Q2. State the formula for the $n^{th}$ term $(a_n)$ of an Arithmetic Progression.

The formula for the nth term of an A.P. is given as:

a_n = a + (n - 1)d,

where ‘a’ is the first term and ‘d’ is the common difference.

Q.3 Define the sum of the first n terms $(S_n)$ of an Arithmetic Progression.

Ans: The sum of the first n terms of an A.P. is denoted as Sn and can be calculated using the formula: Sn = (n/2) [2a + (n – 1) d], where ‘a’ is the first term and ‘d’ is the common difference.

Q4. How do I determine the number of solutions of a quadratic equation without solving it?

Ans: Yes, the common difference in an A.P. can be negative. It means that the sequence is decreasing, and each term is obtained by subtracting the absolute value of ‘d’ from the previous term.

Q5. In an Arithmetic Progression, what is the condition for finding the $n^{th}$ term $(a_n)$ or the sum of the first n terms $(Sn)$ directly?

Ans: To find the nth term

(a_n)

or the sum of the first n terms

(Sn)

directly, the common difference (d) must be known, and the first term (a) should be given or easily determined.

Chapter Wise Important Questions for CBSE Board Class 10 Maths
Real Numbers
Polynomials
Pair of Linear Equations in Two Variables
Quadratic Equations
Arithmetic Progressions
Triangles
Coordinate Geometry
Introduction to Trigonometry
Some Applications of Trigonometry
Circles
Areas Related to Circles
Surface Areas and Volumes
Statistics
Probability

Class 10 Math Chapter 5 Arithmetic Progressions Important Questions

Introduction

What is an Arithmetic Progression (A.P.)?

Class 10 Arithmetic Progressions Important Questions

Q 1. The sum of first 20 odd natural numbers is :

Options

(a) 100(b) 210(c) 400(d) 420

Q 2. The number of multiples of 4 that lie between 10 and 250 is

Options

(a) 62(b) 60(c) 59(d) 55

Q 3. If the sum of n terms of an A.P. is S_n = 3n^2 + 5n, then write its common difference.

Q 4. Which term of the progression

20, 19 \dfrac{1}{4}, 18\dfrac{1}{2}, 17\dfrac{3}{4}, ….. is the first negative term?

Q 5. The houses in a row are numbered consecutively from 1 to 49. Show that there exists a value of X such that the sum of numbers of houses proceeding the house numbered X is equal to sum of the numbers of houses following X.

CBSE Class 10 Maths Chapter wise Important Questions

Conclusion

Frequently Asked Questions

Q1. How many solutions can a quadratic equation have?

Q2. State the formula for the n^{th} term (a_n) of an Arithmetic Progression.

Q.3 Define the sum of the first n terms (S_n) of an Arithmetic Progression.

Q4. How do I determine the number of solutions of a quadratic equation without solving it?

Q5. In an Arithmetic Progression, what is the condition for finding the n^{th} term (a_n) or the sum of the first n terms (Sn) directly?

Class 10 Math Chapter 5
Arithmetic Progressions
Important Questions

(a) 100
(b) 210
(c) 400
(d) 420

(a) 62
(b) 60
(c) 59
(d) 55

Q 3. If the sum of n terms of an A.P. is $S_n = 3n^2 + 5n,$ then write its common difference.

$20, 19 \dfrac{1}{4}, 18\dfrac{1}{2}, 17\dfrac{3}{4},$ ….. is the first negative term?

Q2. State the formula for the $n^{th}$ term $(a_n)$ of an Arithmetic Progression.

Q.3 Define the sum of the first n terms $(S_n)$ of an Arithmetic Progression.

Q5. In an Arithmetic Progression, what is the condition for finding the $n^{th}$ term $(a_n)$ or the sum of the first n terms $(Sn)$ directly?