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

(i) 2, 4, 8, 16, . . . 

Here a = 2 ,

d= 4-2 = 2

   = 8- 4 = 4

The common difference between the consecutive term is not the same.

Therefore, 2, 4, 8, 16, . . .  does not form an AP

(ii) 2, $\frac{5}{2} ,3 , \frac{7}{2}$

Here a = 2 ,

d= $\frac{5}{2}$-2 = 1/2

   = 3- $\frac{5}{2}$ = 1/2

The common difference between the consecutive term same.

Common difference = d = 1/2

The next 3 terms will be, 

4, 9/2 , 5

Therefore, 2, 4, 8, 16, . . .  does not form an AP

(iii) – 1.2, – 3.2, – 5.2, – 7.2, . . . 

Here, 

a = -1.2

d = -3.2 - (-1.2) = -3.2 + 1.2 = - 2

  = a₃ - a₂ = - 5.2 - (-3.2) = - 5.2 + 3.2 = - 2

Here the common difference is same ,so the series is an AP d = -2

The next three terms are : -7.2 -2 = -9.2 , -9.2 -2 = -11.2, -11.2 - 2 = -13.2

The next three terms are : - 9.2, - 11.2, - 13.2

(iv) – 10, – 6, – 2, 2, . . .

Here a= -10

d= -6 + 10 = 4

 = a₃ - a₂ = -2 + 6 = 4

Here the common difference is the same, so the series is an AP  d = 4

The next three terms are : 2+4 = 6, 6+4 = 10, 10+4 = 14

The next three terms are: 6, 10, 14

(v) 3, 3 +$\sqrt{2}$ , 3 + 2$\sqrt{2}$,   3 + 3$\sqrt{2}$, ..

Here a= 3

d= 3 +$\sqrt{2}$ - 3 = $\sqrt{2}$

 = a₃ - a₂ = $\sqrt{2}$

Here the common difference is the same, so the series is an AP , d = $\sqrt{2}$

The next three terms are = 3 + 3$\sqrt{2}$ +$\sqrt{2}$ , 3+4$\sqrt{2}$ + $\sqrt{2}$, 3+5$\sqrt{2}$+$\sqrt{2}$

The next three terms are : 3 + 4√2, 3 + 5√2, 3 + 6√2

(vi) 0.2, 0.22, 0.222, 0.2222, . . .

Initial term: a1 = 0.2

Here d = a₂ - a₁

= 0.22 - 0.2

= 0.02

Common deviation d = a₃ - a₂

= 0.222 - 0.220

= 0.002

The common difference is not the same so the given series is not an AP

(vii) 0, – 4, – 8, –12, . . .

First term a₁ = 0

 d is = a₂ - a₁ = - 4 - 0 = - 4

 d is = a₃ - a₂ = - 8 - (- 4) = - 8 + 4 = - 4

Since the common differences is same , it forms an AP.

The next three terms of AP are , 0 + 4(- 4) = - 16,  0 + 5(- 4) = - 20, 0 + 6(- 4) = - 24

 common difference of - 4. The next three terms are -16, -20, -24.

viii) - 1/2, - 1/2, - 1/2, - 1/2,....

First term a₁ = - 1/2

 d = a₂ - a₁

= - 1/2 - (- 1/2)

= - 1/2 + 1/2

= 0

 d = a₃ - a₂

= - 1/2 - (- 1/2)

= - 1/2 + 1/2

= 0

Since the common difference is the same, the list of numbers forms an AP.

The next three terms are  - 1/2 + 4 (0) = - 1/2,  - 1/2 + 5 (0) = - 1/2, - 1/2 + 6 (0) = - 1/2

 common difference d = 0. The next three terms are - 1/2, - 1/2, - 1/2. 

ix) 1, 3, 9, 27, ...

First term a₁ = 1

 d = a₂ - a₁ = 3 - 1 = 2

 d = a₃ - a₂ = 9 - 3 = 6

Since, the common difference is not the same the given list of numbers does not form an AP.

1, 3, 9, 27 numbers do not form an AP.

x) a, 2a, 3a, 4a,.....

First term a₁ = a

 d = a₂ - a₁

= 2a - a = a

 d = a₃ - a₂

= 3a - 2a = a

Since the common difference is the same the given series forms an AP.

The next terms are : a + 4a = 5a, a + 5a = 6a, a + 6a = 7a

 Common difference d = a. The next three terms are 5a, 6a, 7a.

xi) a, a², a³, a⁴......

First term a₁ = a

d = a₂ - a₁

= a² - a = a (a - 1)

 d = a₃ - a₂

= a³ - a² = a² (a - 1)

Since the common difference is not the same the given list of numbers does not form an AP.

xii) √2, √8, √18, √32......

First term a₁ = √2

 d = a₂ - a₁

= √8 - √2

= 2√2 - √2 = √2

 d = a₃ - a₂

= √18 - √8

= 3√2 - 2√2 = √2

Since the common difference is the same the given numbers form an AP.

The next numbers are as follows: √25 × 2 = √50,  √36 × 2 = √72,   √49 × 2 = √98

 Common difference of √2. The next three terms are √50, √72, √98

xiii) √3, √6, √9, √12, ...

First term a₁ = √3

 d = a₂ - a₁

= √6 - √3

= √3 × 2 - √3

= √3 (√2 - 1)

 d = a₃ - a₂ = √9 - √6

= √3 × 3 - √3 × 2

= √3 (√3 - √2)

Since The common difference is not the same the given list of numbers does not form an AP.

xiv) 1², 3², 5², 7², ...

First term (a) = 1²

 d = a₂ - a₁ = 9 - 1 = 8

 d = a₃ - a₂ = 25 - 9 = 16

Since the common difference is not the same the given list of numbers does not form an AP.

xv) 1², 5², 7², 73, ...

First term a₁ = 1²

 d = a₂ - a₁ = 25 - 1 = 24

 d = a₃ - a₂ = 49 - 25 = 24

Since the common difference is the same they form an AP

The next three terms are : 1 + 96 = 97,  1 + 120 = 121,  1 + 144 = 145

 Common difference of 24. The next three terms are 97, 121, and 145.