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

Case Study

A man writes a letter to four of his friends. He asks each one of them to copy the letter and mail it to four different persons with the instruction that they move the chain similarly. Assume that the chain is not broken and it costs ₹4 to mail one letter.

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

1.The number of letters mailed in the 6th set of letters is :
(a) 2048
(b) 8192
(c) 4096
(d) 1024

2.The amount spent on the postage of 6th set of letters is :
(a) ₹ 16,384
(b) ₹ 8,192
(c) ₹ 32,768
(d) ₹ 4,096

3.The total number of letters mailed till the 5th set of letters is :
(a) 1024
(b) 4092
(c) 1236
(d) 1364

4.The amount spent on the postage till the 5th set of letters is :
(a) ₹ 16,368
(b) ₹ 4,096
(c) ₹ 5,456
(d) ₹ 4,944

5.The amount spent on the postage till the nth set of letters is :
(a) ₹ $\Big(\dfrac{4}{3}(4^n - 4)\Big)$

(b) ₹ $\Big(\dfrac{16}{3}(4^n - 1)\Big)$

(c) ₹ $\Big(\dfrac{2}{3}(4^n - 1)\Big)$

(d) ₹ $\Big(16(4^n - 1)\Big)$

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$2}{a1} = \dfrac{a3}{a2} = \dfrac{a4}{a3} = constant known as the common ratio.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

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. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false