Maharashtra Board Class 11 Maths Part 2 Chapter 2 Sequences and Series Miscellaneous Exercise 2 Solution

(I) Select the correct answer from the given alternative:

Question 1.
The common ratio for the G.P. 0.12, 0.24, 0.48, is

(A) 0.12
(B) 0.2
(C) 0.02
(D) 2
Answer:
(D) 2

Question 2.
word image 22467 1

Answer:
(C) -128
Hint:
word image 22467 2

Question 3.

(A) 3
(B) 2
(C) 1
(D) -1
Answer:

(A) 3
Hint:
word image 22467 4

Question 4.
Which term of the geometric progression 1, 2, 4, 8, ….. is 2048.
(A) 10th
(B) 11th
(C) 12th
(D) 13th
Answer:

(C) 12th
Hint:
Here, a = 1, r = 2
word image 22467 5

Question 5.
If the common ratio of the G.P. is 5, the 5th term is 1875, the first term is
(A) 3
(B) 5
(C) 15
(D) -5
Answer:

(A) 3

Question 6.
The sum of 3 terms of a G.P. is 21/4 and their product is 1, then the common ratio is
(A) 1
(B) 2
(C) 4
(D) 8
Answer:

(C) 4
Hint:
word image 22467 6

Question 7.
word image 22467 7

word image 22467 8

Question 9.
Which of the following is not true, where A, G, H are the AM, GM, HM of a and b respectively, (a, b > 0)

Answer:

(D) A = GH

Question 10.
The G.M. of two numbers exceeds their H.M. by 6/5, the A.M. exceeds G.M. by 3/2 the two numbers are

Answer:

(C) 3, 12
Hint:
word image 22467 11
word image 22467 12

(II) Answer the following:

Question 1.
In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term.
Solution:

word image 22467 13

Question 2.
Find the sum of the first 5 terms of the G.P. whose first term is 1 and the common ratio is 2/3.
Solution:

word image 22467 14

Question 3.

Solution:

word image 22467 16

Question 4.

word image 22467 18

Question 5.
Find three numbers in G.P. such that their sum is 35 and their product is 1000.
Solution:

word image 22467 19
Substituting the value of a in (i), we get
word image 22467 20
Hence, the three numbers in G.P. are 20, 10, 5, or 5, 10, 20.

Question 6.
Find five numbers in G.P. such that their product is 243 and the sum of the second and fourth numbers is 10.
Solution:

word image 22467 21
According to the given condition,
word image 22467 22
word image 22467 23

Question 7.
For a sequence, Sn = 4( – 1), verify that the sequence is a G.P.
Solution:

word image 22467 24
∴ The given sequence is a G.P.

Question 8.
Find 2 + 22 + 222 + 2222 + … upto n terms.
Solution:

word image 22467 26

Question 9.
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666,…
Solution:

word image 22467 27
word image 22467 28

Question 10.

Solution:
word image 22467 30

Question 11.

Solution:

word image 22467 32

Question 12.

Solution:

We know that
word image 22467 34

Question 13.

Solution:
word image 22467 36

Question 14.
Find 2 × 6 + 4 × 9 + 6 × 12 + ….. upto n terms.
Solution:

2, 4, 6, ….. are in A.P.
∴ rth term = 2 + (r – 1) 2 = 2r
6, 9, 12, ….. are in A.P.
∴ rth term = 6 + (r – 1)(3) = (3r + 3)
∴ 2 × 6 + 4 × 9 + 6 × 12 + ….. to n terms
word image 22467 37
= n(n + 1) [2n + 1 + 3]
= 2n(n + 1)(n + 2)

Question 15.
Find 2 × 5 × 8 + 4 × 7 × 10 + 6 × 9 × 12 + …… upto n terms.
Solution:

2, 4, 6,… are in A.P.
∴ rth term = 2 + (r – 1) 2 = 2r
5, 7, 9, … are in A.P.
∴ rth term = 5 + (r – 1) (2) = (2r + 3)
8, 10, 12, … are in A.P.
∴ rth term = 8 + (r – 1) (2) = (2r + 6)
2 × 5 × 8 + 4 × 7 × 10 + 6 × 9 × 12 + ….. to n terms
word image 22467 38
= 2n (n + 1) [n(n + 1) + 3(2n + 1) + 9]
= 2n (n + 1)( + 7n + 12)
= 2n (n + 1) (n + 3) (n + 4)

Question 16.

Solution:
word image 22467 40
word image 22467 41

Question 17.

Solution:

word image 22467 43

Question 18.

Solution:

word image 22467 45

Question 19.

Solution:

word image 22467 47

Question 20.

Solution:
word image 22467 49
word image 22467 50

Question 21.
For a G.P. if  = 7,  = 1575, find a.
Solution:

word image 22467 51

Question 22.

Solution:
word image 22467 53

Question 23.

Solution:

word image 22467 55

Question 24.
Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.
Solution:
Since k – 1, k, k + 2 are consecutive terms of a G.P.,
word image 22467 56
k – 2 = 0
∴ k = 2

Question 25.
If for a G.P. first term is (27 and the seventh term is (8, find .
Solution:
word image 22467 57

Question 26.
If pth, qth and rth terms of a G.P. are x, y, z respectively. Find the value of  .
Solution:

Let a be the first term and R be the common ratio of the G.P.
word image 22467 58

Question 27.
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
Solution:

Let the required numbers be G1 and G2.
word image 22467 59
∴ For the resulting sequence to be in G.P. we need to insert numbers 10 and 20.

Question 28.

Solution:
word image 22467 61

Question 29.

Solution:
a, b, c are in G.P.
∴  = ac
a + 2bx + c = 0 becomes
word image 22467 63

Question 30.

Solution:

p, q, r, s are in G.P.
word image 22467 65

Question 31.
word image 22467 66
i.e., we have to show
word image 22467 67

Question 32.
Find the coefficient  in the expression of  using series expansion.
Solution:

word image 22467 68

Question 33.

Solution:
word image 22467 70
word image 22467 71