Question 1.
Check whether the following sequences are G.P. If so, write tn.
(i) 2, 6, 18, 54, ……
Solution:
(ii) 1, -5, 25, -125, ………
Solution:
Solution:
(iv) 3, 4, 5, 6, ……
Solution:
(v) 7, 14, 21, 28, ……
Solution:
Question 2.
For the G.P.
Solution:
Solution:
(iii) If r = -3 and = 1701, find a.
Solution:
Solution:
Question 3.
Which term of the G. P. 5, 25, 125, 625, …… is ?
Solution:
Question 4.
Solution:
Question 5.
Solution:
Question 6.
Find three numbers in G. P. such that their sum is 21 and the sum of their squares is 189.
Solution:
Let the three numbers in G. P. be , a, ar.
According to the given conditions,
When a = 6, r = 2,
= 3, a = 6, ar = 12
Hence, the three numbers in G.P. are 12, 6, 3 or 3, 6, 12.
Check:
If sum of the three numbers is 21 and sum of their squares is 189, then our answer is correct.
Sum of the numbers = 12 + 6 + 3 = 21
Sum of the squares of the numbers =
= 144 + 36 + 9
= 189
Thus, our answer is correct.
Question 7.
Find four numbers in G. P. such that the sum of the middle two numbers is 10/3 and their product is 1.
Solution:
According to the given conditions,
Question 8.
Find five numbers in G. P. such that their product is 1024 and the fifth term is square of the third term.
Solution:
Let the five numbers in G. P. be
According to the given conditions,
Hence, the five numbers in G.P. are
1, 2, 4, 8, 16 or 1, -2, 4, -8, 16.
Question 9.
The fifth term of a G. P. is x, the eighth term of a G.P. is y and the eleventh term of a G.P. is z, verify whether = xz.
Solution:
Question 10.
If p, q, r, s are in G.P., show that p + q, q + r, r + s are also in G. P.
Solution:
∴ p + q, q + r, r + s are in G.P.
Question 11.
The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there at the end of the 5th hour?
Solution:
Question 12.
A ball is dropped from a height of 80 ft. The ball is such that it rebounds of the height it has fallen. How high does the ball rebound on the 6th bounce? How high does the ball rebound on the nth bounce?
Solution:
Since the ball rebounds of the height it has fallen, the height in successive bounce is in G.P.
1st height in the bounce = 80 × 3/4
Question 13.
The numbers 3, x and x + 6 are in G. P. Find
(i) x
(ii) 20th term
(iii) nth term.
Solution:
(i) 3, x and x + 6 are in G. P.
(x – 6) (x + 3) = 0
x = 6, -3
Question 14.
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning, write down the number of mosquitoes after
(i) 3 years
(ii) 10 years
(iii) n years
Solution:
(iii) Number of mosquitoes after n years = 200 (1.1+
Question 15.
The numbers x – 6, 2x and are in G. P. Find
(i) x
(ii) 1st term
(iii) nth term
Solution:
(ii) = x – 6 = 10 – 6 = 4
