Which of the following are AP? (i) 1, 3, 9, 27, …….. (ii) a, 2a, 3a, 4a, ………..

Question

Which of the following are AP? If they form an AP, find the common difference ’d’ and write three more terms.

(i) 1, 3, 9, 27, …….. 

(ii) a, 2a, 3a, 4a, ……….. 

(iii) a, a² , a³ , a⁴ , ……….. 

(iv) √2, √8, √18, √32 ...

(v) √3, √6, √9, √12

(vi) 1¹, 3², 5², 7², 

(vii) 1¹, 5², 7², 7³, ……….

Answer 1

(i) 1, 3, 9, 27 

d = a₂ – a₁ = 3—1 

d = 2 d = a₃ – a₂ 

d = 6 

Here ‘d’ not is constant 

∴ Given set of numbers do not form an A.P. 

(ii) a, 2a, 3a, 4a 

a₁ = a 

a₂ = 2a 

d = a₂ – a₁ = 2a - a 

d = a 

d = a₃ – a₂ = 3a - 2a 

d = a 

Here d’ is constant. 

∴ Given set of numbers form an A.P. 

Further three terms are : 

4a + a = 5a 

5a + a = 6a 

6a + a = 7a 

∴ Next three terms are: 5a, 6a, 7a. 

(iii) a, a² , a³ , a⁴ 

a₁ = a. a₂ = a₂ 

d = a₂ – a₁ = a₂ – a 

d= a(a – 1) 

d = a₃ – a₂ = a³ – a² 

d = a² (a – 1) 

Here ‘d is not constant. 

∴ Given set of numbers do not form an A.P.

(iv)

Here d’ is constant. 

∴ Given set of numbers form an A.P. 

Succeeding three terms are :

Here ‘d is not constant. 

∴ Given set of numbers do not form an A.P. 

(vi) 1¹,3²,5², 7², ………… 

1, 9, 25, 49, …………. 

a₁ = 1, a₂ = 9 

d = a₂ – a₁ = 9 – 1 

d = 8 

d = a₃ – a₂ = 25 – 9 

d = 16 

Here ‘d’ is not constant, 

∴ Given set of numbers do not form an A.P. 

(vii) 1¹ , 5² , 7² , 73, ………. 

1, 25. 49, 73, ………. 

a₁ = 1. a₂ = 25, a₃ = 49, a₄ = 73 

d = a₂ – a₁ = 25 – 1 

d = 24 

d = a₃ – a₂ = 49 – 25 

d = 24 

d = a₄ – a₃ = 73 – 49 

d = 24 

Here ‘d’ is constant. 

∴ Given set of numbers form an A.P. 

Succeeding three terms are: 

73 + 24 = 97 

97 + 24 = 121 

121 + 24 = 145 

∴ Next three terms are: 97, 121, 145.