Class 10 Maths Chapter 5 - Arithmetic Progressions | INSTALEARN

Question 1

Assertion (A): In an arithmetic progression, the difference between any two consecutive terms is constant.
Reason (R): The formula for the $n$ -th term of an arithmetic progression is $a_n = a + (n - 1)d$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The difference between consecutive terms is the common difference $d$ , which is used in the formula.

Question 2

Assertion (A): If the first term of an arithmetic progression is 2 and the common difference is 3, then the 10th term is 29.
Reason (R): The $n$ -th term of an arithmetic progression is given by $a_n = a + (n - 1)d$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Using the formula, $a_{10} = 2 + (10 - 1) \cdot 3 = 29$ .

Question 3

Assertion (A): The sequence $5, 10, 20, 40, \dots$ is an arithmetic progression.
Reason (R): An arithmetic progression has a constant common difference.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: This sequence is a geometric progression, not an arithmetic progression, as the ratio between terms is constant, not the difference.

Question 4

Assertion (A): In an arithmetic progression with a common difference of zero, all terms are equal.
Reason (R): The $n$ -th term of an arithmetic progression with zero common difference is $a_n = a$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: With a common difference of zero, each term remains the same as the first term.

Question 5

Assertion (A): The sum of the first 10 terms of the arithmetic progression $2, 5, 8, \dots$ is 155.
Reason (R): The sum of the first $n$ terms of an arithmetic progression is $S_n = \frac{n}{2} \left(2a + (n - 1)d\right)$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: Using the formula, $S_{10} = \frac{10}{2} \left(2 \times 2 + 9 \times 3\right) = 185$ .

Question 6

Assertion (A): The sum of the first $n$ natural numbers is an arithmetic progression.
Reason (R): An arithmetic progression has a constant common difference.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: B) Both A and R are true, but R is not the correct explanation of A.
Explanation: The sum itself is not an arithmetic progression; however, the sequence of natural numbers is an arithmetic progression.

Question 7

Assertion (A): For an arithmetic progression, if the first term is doubled and the common difference is halved, the sequence remains an arithmetic progression.
Reason (R): Changing the first term or common difference does not affect the type of progression.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The sequence is still an arithmetic progression with a new first term and common difference.

Question 8

Assertion (A): The sequence $1, 4, 7, 10, \dots$ has a common difference of 4.
Reason (R): The common difference is the result of subtracting the first term from the second term.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: C) A is true, but R is false.
Explanation: The correct common difference is 3 (not 4), and it is obtained by subtracting consecutive terms.

Question 9

Assertion (A): The 12th term of an arithmetic progression with a first term of 5 and a common difference of 2 is 29.
Reason (R): The $n$ -th term of an arithmetic progression is calculated as $a_n = a + (n - 1)d$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: $a_{12} = 5 + 11 \cdot 2 = 29$

Question 10

Assertion (A): An arithmetic progression can have both positive and negative terms.
Reason (R): The common difference can be either positive or negative.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The common difference dictates whether terms increase or decrease, so terms can be either positive or negative.

Question 11

Assertion (A): The sum of the first five terms of an arithmetic progression is equal to five times the third term.
Reason (R): In an arithmetic progression, each term is equal to the previous term plus the common difference.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: C) A is true, but R is false.
Explanation: This assertion is true, but it is a specific case. The reason provided does not explain why the sum equals five times the third term.

Question 12

Assertion (A): In any arithmetic progression, the $n$ -th term can be zero.
Reason (R): The formula for the $n$ -th term is $a_n = a + (n - 1)d$ , which can be set to zero for specific values of $a$ , $d$ , and $n$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Setting $a + (n - 1)d = 0$ can yield $n$ , so it is possible for some terms to be zero depending on $a$ and $d$ .

Question 13

Assertion (A): If the 5th term of an arithmetic progression is 10, then the 10th term is 20.
Reason (R): The difference between terms in an arithmetic progression is always the same.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: This assertion does not hold true because we lack information about the common difference. The common difference could yield a different 10th term.

Question 14

Assertion (A): The sum of the first $n$ odd numbers is equal to $n^2$ .
Reason (R): The sequence of odd numbers is an arithmetic progression with a common difference of 2.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: B) Both A and R are true, but R is not the correct explanation of A.
Explanation: While the sequence is an arithmetic progression, the sum of $n$ odd numbers being $n^2$ is a specific result unrelated to the common difference.

Question 15

Assertion (A): If the first term of an arithmetic progression is 1 and the common difference is 1, the $n$ -th term is $n$ .
Reason (R): In an arithmetic progression, the $n$ -th term formula is $a_n = a + (n - 1)d$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Substituting the values into the formula, $a_n = 1 + (n - 1) \cdot 1 = n$ .

Question 16

Assertion (A): An arithmetic progression with a positive common difference has a minimum term.
Reason (R): An arithmetic progression with a positive common difference increases indefinitely as $n$ increases.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: A sequence with a positive common difference does not have a minimum term as it grows indefinitely.

Question 17

Assertion (A): If the common difference of an arithmetic progression is negative, all terms will eventually become negative.
Reason (R): In an arithmetic progression, terms decrease when the common difference is negative.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: It depends on the first term. If the first term is large enough, it may take many terms to become negative or never be negative.

Question 18

Assertion (A): If the common difference of an arithmetic progression is zero, then the sequence is constant.
Reason (R): A common difference of zero means that each term is equal to the previous term.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Each term is indeed equal to the previous one, making the sequence constant.

Question 19

Assertion (A): The sum of an arithmetic progression with an infinite number of terms is always infinite.
Reason (R): The sum of an arithmetic progression with $n$ terms is given by $S_n = \frac{n}{2} \left(2a + (n - 1)d\right)$ .

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Since $n$ is infinite, the sum will diverge towards infinity.

Question 20

Assertion (A): An arithmetic progression can never be a constant sequence.
Reason (R): The common difference in an arithmetic progression is always non-zero.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: A constant sequence is an arithmetic progression with a common difference of zero.

Question 21

Assertion (A): In an arithmetic progression, if the first term is positive and the common difference is negative, the terms will eventually become negative.
Reason (R): With a negative common difference, the terms decrease as $n$ increases.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: As the common difference is negative, the terms will keep decreasing, eventually crossing zero and becoming negative.

Question 22

Assertion (A): The arithmetic mean of two numbers is the middle term of an arithmetic progression.
Reason (R): In an arithmetic progression, the middle term is equal to the average of the first and last terms.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: The middle term is indeed the arithmetic mean of the two numbers, which is why both are correct and R explains A.

Question 23

Assertion (A): If three numbers are in an arithmetic progression, the middle term is twice the average of the three numbers.
Reason (R): The sum of the three terms in an arithmetic progression is three times the middle term.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: The middle term is actually the average of the three terms, not twice the average.

Question 24

Assertion (A): If all terms of an arithmetic progression are positive, the common difference must also be positive.
Reason (R): The common difference in an arithmetic progression determines whether the sequence increases or decreases.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: D) A is false, but R is true.
Explanation: The common difference could be zero for the sequence to stay constant, or very small and negative, keeping terms positive for a certain number of terms.

Question 25

Assertion (A): The sum of an infinite arithmetic progression with a non-zero common difference is always infinite.
Reason (R): An infinite sequence with a common difference continues indefinitely, with the sum diverging based on the terms.

A) Both A and R are true, and R is the correct explanation of A.
B) Both A and R are true, but R is not the correct explanation of A.
C) A is true, but R is false.
D) A is false, but R is true.
Answer: A) Both A and R are true, and R is the correct explanation of A.
Explanation: Since the number of terms is infinite, the sum will diverge as long as the common difference is non-zero.

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Class 10 Maths Chapter 5 - Arithmetic Progressions