Arithmetic Progressions – Assertion and Reason Questions

Q 1 :

Assertion (A): The common difference of AP 5, 4, 3, 2, … is –1.

Reason (R): The constant difference between any two consecutive terms of an AP is commonly known as common difference of the AP.

Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false
A is false, but R is true.

1

2

3

4

(1)

We have AP 5, 4, 3, 2, … [common difference = 4 – 5 = 3 – 4 = –1]

∴ Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

Q 2 :

Assertion (A): 5, 10, 15 are three consecutive terms of an AP.

Reason (R): If a, b, c are three consecutive terms of an AP, then 2b = a + c.

Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false.
A is false, but R is true

1

2

3

4

(1)

5, 10, 15 are in AP.
⇒ 10 – 5 = 15 – 10 = 5 (difference is same)

If a, b, c are three consecutive terms of an AP, then
b – a = c – b ⇒ 2b = a + c

So, Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

Q 3 :

Assertion (A): Common difference of an AP whose nth term is given by $a ₙ = 4 n - 70 i s 4$

Reason (R): $d = a ₙ - a ₙ ₋ ₁$

Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A
A is true, but R is false.
A is false, but R is true.

1

2

3

4

(1)

$G i v e n n ᵗ ʰ t e r m o f a n A P i s a ₙ = 4 n - 70$

$d = a ₙ - a ₙ ₋ ₁ = (4 n - 70) - 4 (n - 1) + 70$

$= 4 n - 70 - 4 n + 4 + 70 = 4$

Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

Q 4 :

Assertion (A): a, b, c are in AP if and only if 2b = a + c.

Reason (R): The sum of first n odd natural numbers is n².

Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A
A is true, but R is false.
A is false, but R is true.

1

2

3

4

(2)

Since a, b, c are in AP.

$∴ b - a = c - b \Rightarrow 2 b = a + c A l s o, s u m o f f i r s t n o d d n a t u r a l n u m b e r s = 1 + 3 + 5 + . . . w h i c h a r e i n A P w i t h a = 1, d = 3 - 1 = 2 ∴ S ₙ = n / 2 (2 a + (n - 1) d) = n / 2 (2 \times 1 + (n - 1) \times 2) = n (1 + n - 1) = n ²$

Q 5 :

Assertion (A): Sum of first 15 terms of 2 + 5 + 8 … is 345.

Reason (R): Sum of first n terms in an AP is given by the formula:

$S_{n} = n / 2 [2 a + (n - 1) d]$

Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false.
A is false, but R is true.

1

2

3

4

(1)

We have AP = 2 + 5 + 8 + …

Here, a = 2, d = 5 – 2 = 3, n = 15

$S_{15} = 15 / 2 (2 \times 2 + 14 \times 3) = 15 / 2 (4 + 42) = 15 / 2 \times 46 = 15 \times 23 = 345$

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Q 6 :

Assertion (A): The sum of the series with the nth term, $a_{n} = 9 - 5 n$ is –465 when number of terms, n = 15.

Reason (R): The sum of first n terms of an AP is given by $S_{n} = n / 2 [2 a + (n - 1) d]$

Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A
A is true, but R is false
A is false, but R is true.

1

2

3

4

(1)

$G i v e n n t h t e r m a_{n} = 9 - 5 n F i r s t t e r m, a_{1} = 9 - 5 \times 1 = 4 \Rightarrow a = 4 L a s t t e r m, a_{15} = 9 - 5 \times 15 = 9 - 75 = - 66 S u m o f t h e s e r i e s : S_{15} = 15 / 2 (a + l) = 15 / 2 (4 - 66) = 15 / 2 (- 62) = 15 \times (- 31) = - 465 A l s o b y t h e f o r m u l a, S_{n} = n / 2 (2 a + (n - 1) d) = n / 2 (a + l)$

Q 7 :

Assertion (A): Sum of first n terms of the AP 3, 13, 23 … is $5 n^{2} - 8 n$ .

Reason (R): The sum of first n terms of an AP is given by $S_{n} = n / 2 [2 a + (n - 1) d]$

Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A
A is true, but R is false
A is false, but R is true.

1

2

3

4

(4)

We have AP 3, 13, 23, … Here,

a = 3, d = 13 – 3 = 10

$∴ S_{n} = \frac{n}{2} (2 a + (n - 1) d) = \frac{n}{2} (2 \times 3 + (n - 1) \times 10) = n (3 + (n - 1) \times 5) \Rightarrow S_{n} = n (5 n - 2) = 5 n^{2} - 2 n$

Topic Question Set

Q 1 :

Assertion (A): The common difference of AP 5, 4, 3, 2, … is –1.

Reason (R): The constant difference between any two consecutive terms of an AP is commonly known as common difference of the AP.

Q 2 :

Assertion (A): 5, 10, 15 are three consecutive terms of an AP.

Reason (R): If a, b, c are three consecutive terms of an AP, then 2b = a + c.

Q 3 :

Assertion (A): Common difference of an AP whose nth term is given by $a ₙ = 4 n - 70 i s 4$

Reason (R): $d = a ₙ - a ₙ ₋ ₁$

Q 4 :

Assertion (A): a, b, c are in AP if and only if 2b = a + c.

Reason (R): The sum of first n odd natural numbers is n².

Q 5 :

Assertion (A): Sum of first 15 terms of 2 + 5 + 8 … is 345.

Reason (R): Sum of first n terms in an AP is given by the formula:

$S_{n} = n / 2 [2 a + (n - 1) d]$

Q 6 :

Assertion (A): The sum of the series with the nth term, $a_{n} = 9 - 5 n$ is –465 when number of terms, n = 15.

Reason (R): The sum of first n terms of an AP is given by $S_{n} = n / 2 [2 a + (n - 1) d]$

Q 7 :

Assertion (A): Sum of first n terms of the AP 3, 13, 23 … is $5 n^{2} - 8 n$ .

Reason (R): The sum of first n terms of an AP is given by $S_{n} = n / 2 [2 a + (n - 1) d]$

Want Us to Email you About Special Offers & Updates?