Question 1

Question 1: Verify whether the following sequences are G.P. If so, write tn. (i) 2, 6, 18, 54, …… (ii) 1, -5, 25, -125, ……. (iii) (iv) 3, 4, 5, 6, …… (v) 7, 14, 21, 28, …..

Accepted Answer

(i) 2, 6, 18, 54, ……. t1 = 2, t2 = 6, t3 = 18, t4 = 54, ….. Here, Since, the ratio of any two consecutive terms is a constant, the given sequence is a geometric progression. Here, a = 2, r = 3 tn= arn-1 ∴ tn = 2(3n-1)

Question 2

Question 2: For the G.P., (i) if r = , a = 9, find t7. (ii) if a = , r = , find t3. (iii) if a = 7, r = -3, find t6. (iv) if a = , t6 = 162, find r.

Accepted Answer

Step-by-step solution provided.

Question 3

Question 3: Which term of the G. P. 5, 25, 125, 625, ….. is 510?

Accepted Answer

Step-by-step solution provided.

Question 4

Question 4: For what values of x, , x, are in G. P.?

Accepted Answer

Step-by-step solution provided.

Question 5

Question 5: If for a sequence, , show that the sequence is a G. P. Find its first term and the common ratio.

Accepted Answer

The sequence (tn) is a G.P., if = constant, for all n ∈ N ∴ the sequence is a G. P. with common ratio First term, t1 =

Question 6

Question 6: Find three numbers in G. P. such that their sum is 21 and sum of their squares is 189.

Accepted Answer

Let the three numbers in G. P. be , a, ar. According to the first condition, ∴ the three numbers are 12, 6, 3 or 3, 6, 12. Check: First condition: 12, 6, 3 are in G.P. with r = 12 + 6 + 3 = 21 Second condition: 122 + 62 + 32 = 144 + 36 + 9 = 189 Thus, both the conditions are satisfied.

Question 7

Question 7: Find four numbers in G. P. such that sum of the middle two numbers is and their product is 1.

Accepted Answer

Let the four numbers in G.P. be . According to the second condition, ∴ a4 = 1 ∴ a = 1 According to the first condition,

Question 8

Question 8: Find five numbers in G. P. such that their product is 1024 and the fifth term is square of the third term.

Accepted Answer

Let the five numbers in G. P. be According to the given conditions, When a = 4, r = -2 = 1, = -2, a = 4, ar = -8, ar2 = 16 ∴ the five numbers in G.P. are 1, 2, 4, 8, 16 or 1, -2, 4, -8, 16.

Question 9

Question 9: The fifth term of a G. P. is x, eighth term of the G. P. is y and eleventh term of the G. P. is z. Verify whether y2 = xz.

Accepted Answer

Step-by-step solution provided.

Question 10

Question 10: If p, q, r, s are in G. P., show that p + q, q + r, r + s are also in G.P.

Accepted Answer

p, q, r, s are in G.P. ∴ p + q, q + r, r + s are in G.P.

Chapter 4 Sequences and Series Ex 4.1 Solutions