Geometric Progression — Assertion-Reason Type Questions

Assertion (A): The nth term of a G.P. is given by T_n = ar^{n − 1}.

Reason (R): A sequence a₁, a₂, a₃, ....... is said to be a G.P. if $\dfrac{a_1}{a_2} = \dfrac{a_2}{a_3} = \dfrac{a_3}{a_4}$ = constant known as the common ratio.

1. Both A and R are true, and R is the correct explanation of A.

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

3. A is true, but R is false.

4. A is false, but R is true.

Answer

The nth term of a G.P. is given by :

T_n = ar^{n - 1}

Assertion (A) is true.

A sequence $a_1, a_2, a_3, \dots$ is said to be a G.P. if $\dfrac{a_2}{a_1} = \dfrac{a_3}{a_2} = \dfrac{a_4}{a_3} = \text{constant}$ known as the common ratio.

Reason (R) is false.

Hence, option 3 is the correct option.

Assertion (A): The 50th term from the end of the G.P. 4, 6, 9, $\dfrac{27}{2}, ......\dfrac{6561}{64}$ is $\dfrac{27}{2}$.

Reason (R): nth term from the end of a G.P. is given by $\dfrac{l}{r^{n-1}}$.

1. Both A and R are true, and R is the correct explanation of A.

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

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Given,

a = 4

n = 50

r = $\dfrac{6}{4} = \dfrac{3}{2}$

l = $\dfrac{6561}{64}$

We know that,

$\text{nth term from end} = \dfrac{l}{r^{n - 1}}$

Substituting values we get,

$\Rightarrow \text{50th term from end} = \dfrac{\dfrac{6561}{64}}{\Big(\dfrac{3}{2}\Big)^{50-1}} \\[1em] = \dfrac{\dfrac{3^8}{2^6}}{\dfrac{3^{49}}{2^{49}}} \\[1em] = \dfrac{3^8}{2^6} \times \dfrac{2^{49}}{3^{49}} \\[1em] = 2^{49 - 6} \times 3 ^{8 - 49} \\[1em] = 2^{43} \times 3^{-41}$

= $\dfrac{2^{43}}{3^{41}}$ ≠ $\dfrac{27}{2}$.

So, Assertion is false.

nth term from the end of a G.P. is given by: $\dfrac{l}{r^{n-1}}$

So, Reason (R) is true.

Hence, option 4 is the correct option.

Assertion (A): The sum of first 10 terms of the G.P. 3, 6, 9, 12, ...... is 3096.

Reason (R): The sum of first n-terms of a G.P. is given by $S_n = \dfrac{a(r^n - 1)}{r - 1}$.

1. Both A and R are true, and R is the correct explanation of A.

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

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Given,

3, 6, 9, 12, …

Since, $\dfrac{6}{3}$ ≠ $\dfrac{9}{6}$

Thus, r is not constant. Hence it is not G.P.

Assertion (A) is false.

The correct formula for the sum of n terms of a G.P.

$S_n = \dfrac{a(r^n - 1)}{r - 1}$ [r > 1]

Reason (R) is true.

Hence, option 4 is the correct option.