RD Sharma Class 11 Solutions Chapter - Geometric Progressions | Mathematics (Maths) Class 11 - Commerce PDF Download

 Page 1


20. Geometric Progressions
Exercise 20.1
1. Question
Show that each one of the following progressions is a G.P. Also, find the common ratio in each case :
i. 4, -2, 1, ,….
ii. 
iii. 
iv. 
Answer
(i) Let a = 4, b = -2, c = 1.
In GP, b
2
=ac
? (-2)
2
 = 4.1
? 4 = 4
Common ratio = r 
(ii) Let , b = -6, c = -54.
In GP, b
2
=ac
? 
? 36 = 36
Common ratio = r 
(iii) Let a = a, 
In GP, b
2
=ac
? 
? 
Common ratio 
(iv) Let 
In GP, b
2
=ac
? 
? 
Page 2


20. Geometric Progressions
Exercise 20.1
1. Question
Show that each one of the following progressions is a G.P. Also, find the common ratio in each case :
i. 4, -2, 1, ,….
ii. 
iii. 
iv. 
Answer
(i) Let a = 4, b = -2, c = 1.
In GP, b
2
=ac
? (-2)
2
 = 4.1
? 4 = 4
Common ratio = r 
(ii) Let , b = -6, c = -54.
In GP, b
2
=ac
? 
? 36 = 36
Common ratio = r 
(iii) Let a = a, 
In GP, b
2
=ac
? 
? 
Common ratio 
(iv) Let 
In GP, b
2
=ac
? 
? 
Common ratio
2. Question
Show that the sequence defined by n ? N is a G.P.
Answer
Put n = 1,2,3,4…
a
1
, a
2
, a
3
, a
4
? 
If a
1
, a
2
, a
3
 …
? (a
2
)
2
 = a
1
.a
3
? 
? 
So, It is GP.
Common ratio 
3 A. Question
Find:
the ninth term of the G.P. 1, 4, 16, 64, ….
Answer
T
n
 = ar
n-1
a = 4, 
? T
9
 = 4.(4
9-1
)
= 4.4
8
= 4
9
? The 9 term is 4
9
3 B. Question
Find:
the 10
th
 term of the G.P. 
Answer
T
n
 = ar
n-1
Page 3


20. Geometric Progressions
Exercise 20.1
1. Question
Show that each one of the following progressions is a G.P. Also, find the common ratio in each case :
i. 4, -2, 1, ,….
ii. 
iii. 
iv. 
Answer
(i) Let a = 4, b = -2, c = 1.
In GP, b
2
=ac
? (-2)
2
 = 4.1
? 4 = 4
Common ratio = r 
(ii) Let , b = -6, c = -54.
In GP, b
2
=ac
? 
? 36 = 36
Common ratio = r 
(iii) Let a = a, 
In GP, b
2
=ac
? 
? 
Common ratio 
(iv) Let 
In GP, b
2
=ac
? 
? 
Common ratio
2. Question
Show that the sequence defined by n ? N is a G.P.
Answer
Put n = 1,2,3,4…
a
1
, a
2
, a
3
, a
4
? 
If a
1
, a
2
, a
3
 …
? (a
2
)
2
 = a
1
.a
3
? 
? 
So, It is GP.
Common ratio 
3 A. Question
Find:
the ninth term of the G.P. 1, 4, 16, 64, ….
Answer
T
n
 = ar
n-1
a = 4, 
? T
9
 = 4.(4
9-1
)
= 4.4
8
= 4
9
? The 9 term is 4
9
3 B. Question
Find:
the 10
th
 term of the G.P. 
Answer
T
n
 = ar
n-1
? T
9
? The 10 term is .
3 C. Question
Find:
the 8
th
 term of the G.P., 0.3, 0.06, 0.012, ….
Answer
T
n
 = ar
n-1
? T
8
? The 10 term is .
3 D. Question
Find:
the 12th term of the G.P. ax, a
5
, x
5
, ….
Answer
T
n
 = ar
n-1
? T
12
= a
41
x
41
? The 12 term is a
41
x
41
.
3 E. Question
Find:
nth term of the G.P. 
Page 4


20. Geometric Progressions
Exercise 20.1
1. Question
Show that each one of the following progressions is a G.P. Also, find the common ratio in each case :
i. 4, -2, 1, ,….
ii. 
iii. 
iv. 
Answer
(i) Let a = 4, b = -2, c = 1.
In GP, b
2
=ac
? (-2)
2
 = 4.1
? 4 = 4
Common ratio = r 
(ii) Let , b = -6, c = -54.
In GP, b
2
=ac
? 
? 36 = 36
Common ratio = r 
(iii) Let a = a, 
In GP, b
2
=ac
? 
? 
Common ratio 
(iv) Let 
In GP, b
2
=ac
? 
? 
Common ratio
2. Question
Show that the sequence defined by n ? N is a G.P.
Answer
Put n = 1,2,3,4…
a
1
, a
2
, a
3
, a
4
? 
If a
1
, a
2
, a
3
 …
? (a
2
)
2
 = a
1
.a
3
? 
? 
So, It is GP.
Common ratio 
3 A. Question
Find:
the ninth term of the G.P. 1, 4, 16, 64, ….
Answer
T
n
 = ar
n-1
a = 4, 
? T
9
 = 4.(4
9-1
)
= 4.4
8
= 4
9
? The 9 term is 4
9
3 B. Question
Find:
the 10
th
 term of the G.P. 
Answer
T
n
 = ar
n-1
? T
9
? The 10 term is .
3 C. Question
Find:
the 8
th
 term of the G.P., 0.3, 0.06, 0.012, ….
Answer
T
n
 = ar
n-1
? T
8
? The 10 term is .
3 D. Question
Find:
the 12th term of the G.P. ax, a
5
, x
5
, ….
Answer
T
n
 = ar
n-1
? T
12
= a
41
x
41
? The 12 term is a
41
x
41
.
3 E. Question
Find:
nth term of the G.P. 
Answer
T
n
 = ar
n-1
? T
n
 = 
? The n term is .
3 F. Question
Find:
the 10
th
 term of the G.P.
Answer
T
n
 = ar
n-1
? T
10
? The 10 term is .
4. Question
Find the 4
th
 term from the end of the G.P. 
Answer
Nth term from the end is given by,
N 
Where l = last term, n = nth term, and r = common ratio.
L = last term = 162
common ratio 
n = 4
? N 
? N 
? 6
Page 5


20. Geometric Progressions
Exercise 20.1
1. Question
Show that each one of the following progressions is a G.P. Also, find the common ratio in each case :
i. 4, -2, 1, ,….
ii. 
iii. 
iv. 
Answer
(i) Let a = 4, b = -2, c = 1.
In GP, b
2
=ac
? (-2)
2
 = 4.1
? 4 = 4
Common ratio = r 
(ii) Let , b = -6, c = -54.
In GP, b
2
=ac
? 
? 36 = 36
Common ratio = r 
(iii) Let a = a, 
In GP, b
2
=ac
? 
? 
Common ratio 
(iv) Let 
In GP, b
2
=ac
? 
? 
Common ratio
2. Question
Show that the sequence defined by n ? N is a G.P.
Answer
Put n = 1,2,3,4…
a
1
, a
2
, a
3
, a
4
? 
If a
1
, a
2
, a
3
 …
? (a
2
)
2
 = a
1
.a
3
? 
? 
So, It is GP.
Common ratio 
3 A. Question
Find:
the ninth term of the G.P. 1, 4, 16, 64, ….
Answer
T
n
 = ar
n-1
a = 4, 
? T
9
 = 4.(4
9-1
)
= 4.4
8
= 4
9
? The 9 term is 4
9
3 B. Question
Find:
the 10
th
 term of the G.P. 
Answer
T
n
 = ar
n-1
? T
9
? The 10 term is .
3 C. Question
Find:
the 8
th
 term of the G.P., 0.3, 0.06, 0.012, ….
Answer
T
n
 = ar
n-1
? T
8
? The 10 term is .
3 D. Question
Find:
the 12th term of the G.P. ax, a
5
, x
5
, ….
Answer
T
n
 = ar
n-1
? T
12
= a
41
x
41
? The 12 term is a
41
x
41
.
3 E. Question
Find:
nth term of the G.P. 
Answer
T
n
 = ar
n-1
? T
n
 = 
? The n term is .
3 F. Question
Find:
the 10
th
 term of the G.P.
Answer
T
n
 = ar
n-1
? T
10
? The 10 term is .
4. Question
Find the 4
th
 term from the end of the G.P. 
Answer
Nth term from the end is given by,
N 
Where l = last term, n = nth term, and r = common ratio.
L = last term = 162
common ratio 
n = 4
? N 
? N 
? 6
? 4
th
 term from last is 6.
5. Question
which term of the progression 0.004, 0.02, 0.1, …. Is 12.5?
Answer
T
n
 = ar
n-1
, T
n
 = 12.5, n=?
? 
? 
? 
? 
? n = 6
6 A. Question
Which term of the G.P. :
… is ?
Answer
T
n
 = ar
n-1
? 
? 
? 
? 
? n = 11
6 B. Question
Which term of the G.P. :
2, 2 v2,4, …. Is 128?
Answer
T
n
 = ar
n-1
?
? 64v2 = (v2)
n