Arithmetic & Geometric Progression — Assertion-Reason Type Question

Given is a sequence of three terms : -1, -5, -9,......

Assertion (A): They are consecutive term of an arithmetic progression.

Reason (R): Difference between two successive term is -3.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given sequence : -1, -5, -9,......

Difference between 2nd term and 1st term,

⇒ a₂ - a₁ = -5 - (-1)

= -5 + 1 = -4

Difference between 3rd term and 2nd term,

⇒ a₃ - a₂ = -9 - (-5)

= -9 + 5 = -4.

Since, difference between consecutive terms is equal.

Thus, the given sequence is in arithmetic progression with common difference -4.

Thus, Assertion (A) is true, but Reason (R) is false.

Hence, option 1 is the correct option.

T_n = 4 - 2n is the nth term of an A.P.

Assertion (A): This A.P. will have all terms negative after 2nd term.

Reason (R): The common difference is negative.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given, T_n = 4 - 2n

⇒ T₁ = 4 - 2 x 1

= 4 - 2 = 2

⇒ T₂ = 4 - 2 x 2

= 4 - 4 = 0

⇒ T₃ = 4 - 2 x 3

= 4 - 6 = -2

⇒ T₄ = 4 - 2 x 4

= 4 - 8 = -4

So, the sequence 2, 0, -2, -4, ..............

After 2nd term, this A.P. will have all terms negative.

So, assertion (A) is true.

Common difference, T_n - T_{n - 1}

= [4 - 2n] - [4 - 2(n - 1)]

= 4 - 2n - [4 - 2n + 2]

= 4 - 2n - 4 + 2n - 2

= -2.

So, reason (R) is true and reason (R) is the correct explanation of assertion (A).

Hence, option 3 is the correct option.

Observe the sequence of the terms : 3, 8, 13, ..............

Assertion (A): 53 is a term of this A.P.

Reason (R): Its first term is 3 and common difference is 5.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given sequence: 3, 8, 13, ..............

First term = 3

Difference between second and first term :

⇒ a₂ - a₁ = 8 - 3 = 5

Difference between third and second term :

⇒ a₃ - a₂ = 13 - 8 = 5

Thus, the above sequence is an A.P. with first term (a) = 3 and common difference (d) = 5.

So, reason (R) is true.

By formula,

a_n = a + (n - 1)d

If 53 is a term of this A.P., then

⇒ 53 = 3 + (n - 1)5

⇒ 53 - 3 = (n - 1)5

⇒ 50 = (n - 1)5

⇒ n - 1 = $\dfrac{50}{5}$

⇒ n - 1 = 10

⇒ n = 10 + 1 = 11.

Thus, 53 is 11th term of the A.P.

So, assertion (A) is true.

Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.

Given x, y, z are in A.P.

Assertion (A): z, y, x are in A.P.

Reason (R): Terms of an A.P. taken in reverse order also forms an A.P.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given x, y, z are in A.P.

Common difference, d = difference of two consecutive terms,

⇒ a₂ - a₁ = y - x

⇒ a₃ - a₂ = z - y

If x, y, z are in A.P., then d will be equal, y - x = z - y

⇒ x - z = 2y ..................(1)

If z, y, x are in A.P.

⇒ a₂ - a₁ = y - z

⇒ a₃ - a₂ = x - y

If z, y, x are in A.P., then d will be equal, y - z = x - y

⇒ -z + x = 2y

⇒ z - x = -2y

From equation (1), we can write that z, y, x are in A.P. with negative d.

So, assertion (A) is true.

From above we can conclude that the term in A.P. taken in reverse order also form an A.P.

So, reason (R) is true. And, reason (R) is the correct explanation of assertion (A).

Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.

Assertion (A): The sum of first 99 natural numbers is 4950.

Reason (R): The sum of first n natural numbers is $\dfrac{n(n + 1)}{2}$.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Let first n natural numbers : 1, 2, 3, ......., n.

The above sequence is an A.P. with first term = 1, common difference = 1 and number of terms = n.

By formula,

Sum of an A.P. = $\dfrac{n}{2}[2a + (n - 1)d]$

Substituting values we get :

$S = \dfrac{n}{2} \times [2 \times 1 + (n - 1) \times 1] \\[1em] = \dfrac{n}{2} \times [2 + n - 1] \\[1em] = \dfrac{n}{2} \times (n + 1).$

So, reason (R) is true.

When n = 99.

The sum of first 99 natural numbers

$= \dfrac{99(99 + 1)}{2}\\[1em] = \dfrac{99 \times 100}{2}\\[1em] = \dfrac{9900}{2}\\[1em] = 4950$

So, assertion (A) is true and, reason (R) is the correct explanation of assertion (A).

Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.