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}{a2} = \dfrac{a2}{a3} = \dfrac{a3}{a4} = 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.

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.

Write the first five terms of the sequence given by $(\sqrt{3})^n$, n ∈ N.

(a) Is the sequence an A.P. or G.P.?

(b) If the sum of its first ten terms is $p(3 + \sqrt{3})$, find the value of p.

Case Study

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, then

Based on the above given information, answer the following questions:

1. The number of bacteria that would be present at the end of 4th hour is :
   (a) 240
   (b) 480
   (c) 960
   (d) 450

2. The number of bacteria that would be present at the end of 10th hour is :
   (a) 30720
   (b) 15360
   (c) 61440
   (d) 27540

3. The number of bacteria that would be present at the end of nth hour is :
   (a) 30 × 2^{(n - 1)}
   (b) 30 × 2^{(n + 1)}
   (c) 30 × 2ⁿ
   (d) none of these

4. What would be the sum of the first 5 terms of the G.P. that is formed by the number of bacteria after the completion of each hour?
   (a) 450
   (b) 930
   (c) 1890
   (d) none of these

5. What would be the sum of the first n terms of the G.P. that is formed by the number of bacteria after the completion of each hour?
   (a) 30 × 2⁽ⁿ⁾ - 1
   (b) 30 × (2^{(n - 1)} - 1)
   (c) 30 × (2^{(n + 1)} - 1)
   (d) 30 × 2ⁿ - 30