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# 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, ...

ii) 2, 5/2 ,3, 7/2, ...

iii) - 1.2, - 3.2, - 5.2, - 7.2, ...

iv) - 10, - 6, - 2, 2, ...

v) 3, 3 + √2, 3 + 2√2, 3 + 3√2, ...

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

vii) 0, - 4, - 8, - 12, ...

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

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

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

xi) a, a², a³, a⁴, ...

xii) √2, √8, √18, √32, ...

xiii) √3, √6, √9, √12, ...

xiv) 1², 3², 5², 7², ...

xv) 1², 5², 7², 73, ...

**Solution:**

General form of an arithmetic progression is a, (a + d), (a + 2d), (a + 3d), ....

Here, a is the first term and d is a common difference.

i) 2, 4, 8, 16......

First term a₁ = 2

Common difference d = a₂ - a₁ = 4 - 2 = 2

Common difference d = a₃ - a₂ = 8 - 4 = 4

(a₃ - a₂) ≠ (a₂ - a₁)

So, 2, 4, 8, 16, ... are not in AP, because the common difference is not equal.

ii) 2, 5/2 ,3, 7/2, ...

First term a₁ = 2

Common difference d = a₂ - a₁ = 5/2 - 2 = (5 - 4)/2 = 1/2

Common difference d = a₃ - a₂ = 3 - 5/2 = (6 - 5)/2 = 1/2

Since a₃ - a₂ = a₂ - a₁.

2, 5/2 ,3, 7/2 forms an AP and common difference is 1/2

The next three terms are:

- Fifth term = a₁ + 4d = 2 + 4 × 1/2 = 2 + 2 = 4
- Sixth term = a₁ + 5d = 2 + 5 × 1/2 = 2 + 5/2 = (4 + 5) / 2 = 9/2
- Seventh term = a + 6d = 2 + 6 × 1/2 = 5

2, 5/2, 3, 7/2, ... forms an AP and the common difference is 1/2. The next three terms are 4, 9/2, 5

iii) - 1.2, - 3.2, - 5.2, - 7.2, ...

First term a₁ = - 1.2

Common difference d = a₂ - a₁ = -3.2 - (-1.2) = -3.2 + 1.2 = - 2

Common difference d = a₃ - a₂ = - 5.2 - (-3.2) = - 5.2 + 3.2 = - 2

Since a₃ - a₂ = a₂ - a₁_{,} it forms an AP.

- Fifth term = a₁ + 4d = - 1.2 + 4(- 2) = -1.2 - 8 = - 9.2
- Sixth term = a₁ + 5d = - 1.2 + 5(- 2) = - 1.2 - 10 = - 11.2
- Seventh term = a₁ + 6d = - 1.2 + 6(- 2) = - 1.2 - 12 = - 13.2

- 1.2, - 3.2, - 5.2, - 7.2, ... forms an AP with common difference - 2. The next three terms of AP are - 9.2, - 11.2, - 13.2

iv) - 10, - 6, - 2, 2, ...

First term a₁ = - 10

Common difference d is a₂ - a₁

= - 6 - (- 10)

= - 6 + 10 = 4

Common difference d is = a₃ - a₂

= - 2 - (- 6)

= - 2 + 6 = 4

Since a₃ - a₂ = a₂ - a₁, - 10, - 6, - 2, 2, ... forms an AP

- Fifth Term: a₁ + 4d = - 10 + 16 = 6
- Sixth Term: a₁ + 5d = - 10 + 20 = 10
- Seventh Term: a₁ + 6d = - 10 + 24 = 14

- 10, -6, - 2, 2 forms an AP with common difference 4 and next three terms are 6, 10, 14.

v) 3, 3 + √2, 3 + 2√2, 3 + 3√2, ...

First term a₁ = 3

Common difference d is = a₂ - a₁

= 3 + √2 - 3

= √2

Common difference d is = a₃_{ }- a₂

= 3 + 2√2 - (3 + √2)

= 3 + 2√2 - 3 - √2

= √2

Since a₃ - a₂ = a₂ - a₁, 3, 3 + √2, 3 + 2√2, 3 + 3√2, ... forms an AP.

So, 3, 3 + √2, 3 + 2√2, 3 + 3√2 forms an AP with common difference 4.

Next three terms are

- Fifth term = a₁ + 4d = 3 + 4 × √2 = 3 + 4√2
- Sixth term = a₁ + 5d = 3 + 5 × √2 = 3 + 5√2
- Seventh term = a₁ + 6d = 3 + 6 × √2 = 3 + 6√2

3, 3 + √2, 3 + 2√2, 3 + 3√2 forms an AP with common difference √2 and next three terms are 3 + 4√2, 3 + 5√2, 3 + 6√2

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

First term a₁ = 0.2

Common difference d = a₂_{ }- a₁

= 0.22 - 0.2

= 0.02

Common difference d = a₃_{ }- a₂

= 0.222 - 0.220

= 0.002

Since (a₃ - a₂) ≠ (a₂ - a₁), 0.2, 0.22, 0.222, 0.2222, ... do not form an AP.

So, the given list of numbers does not form an AP.

vii) 0, - 4, - 8, - 12, ...

First term a₁ = 0

Common difference d is = a₂ - a₁ = - 4 - 0 = - 4

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

Since a₃ - a₂ = a₂ - a₁, it forms an AP.

- Fifth term = a₁ + 4d = 0 + 4(- 4) = - 16
- Sixth term = a₁ + 5d = 0 + 5(- 4) = - 20
- Seventh term = a₁ + 6d = 0 + 6(- 4) = - 24

0, - 4, - 8, - 12 forms an AP with a 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

Common difference d = a₂ - a₁

= - 1/2 - (- 1/2)

= - 1/2 + 1/2

= 0

Common difference d = a₃ - a₂

= - 1/2 - (- 1/2)

= - 1/2 + 1/2

= 0

Since a₃ - a₂ = a₂ - a₁, the list of numbers forms an AP.

- Fifth term = a₁ + 4d = - 1/2 + 4 (0) = - 1/2
- Sixth term = a₁ + 5d = - 1/2 + 5 (0) = - 1/2
- Seventh term = a₁ + 6d = - 1/2 + 6 (0) = - 1/2

- 1/2, - 1/2, - 1/2, - 1/2 forms an AP with a common difference d = 0. The next three terms are - 1/2, - 1/2, - 1/2.

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

First term a₁ = 1

Common difference d = a₂ - a₁ = 3 - 1 = 2

Common difference d = a₃ - a₂ = 9 - 3 = 6

Since a₂ - a₁ ≠ a₃ - a₂, 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

Common difference, d = a₂ - a₁

= 2a - a = a

Common difference, d = a₃ - a₂

= 3a - 2a = a

Since a₃ - a₂ = a₂ - a₁, a, 2a, 3a, 4a, ... forms an AP.

- Fifth term = a₁ + 4d = a + 4a = 5a
- Sixth term = a₁ + 5d = a + 5a = 6a
- Seventh term = a₁ + 6d = a + 6a = 7a

a, 2a, 3a, 4a forms an AP with a common difference d = a. The next three terms are 5a, 6a, 7a.

xi) a, a^{2}, a^{3}, a^{4}......

First term a₁ = a

Common difference, = a₂ - a₁

= a^{2 }- a = a (a - 1)

Common difference, d = a₃ - a₂

= a^{3 }- a^{2} = a^{2} (a - 1)

Since a₂ - a₁ ≠ a₃ - a₂, the given list of numbers does not form an AP.

xii) √2, √8, √18, √32......

First term a₁ = √2

Common difference, d = a₂ - a₁

= √8 - √2

= 2√2 - √2 = √2

Common difference d = a₃_{ }- a₂

= √18 - √8

= 3√2 - 2√2 = √2

Since a₂ - a₁ = a₃ - a₂, the given numbers form an AP.

- Fifth term = a₁ + 4d = √2 + 4√2 = 5√2 = √25 × 2 = √50
- Sixth term = a₁ + 5d = √2 + 5√2 = 6√2 = √36 × 2 = √72
- Seventh term = a₁ + 6d = √2 + 6√2 = 7√2 = √49 × 2 = √98

√2, √8, √18, √32 forms an AP with a common difference of √2. The next three terms are √50, √72, √98

xiii) √3, √6, √9, √12, ...

First term a₁ = √3

Common difference d = a₂ - a₁

= √6 - √3

= √3 × 2 - √3

= √3 (√2 - 1)

Common difference d = a₃ - a₂ = √9 - √6

= √3 × 3 - √3 × 2

= √3 (√3 - √2)

Since a₂ - a₁ ≠ a₃ - a₂, the given list of numbers does not form an AP.

xiv) 1², 3², 5², 7², ...

First tem (a) = 1²

Common difference, d = a₂ - a₁ = 9 - 1 = 8

Common difference, d = a₃ - a₂ = 25 - 9 = 16

Since a₂ - a₁ ≠ a₃ - a₂, the given list of numbers does not form an AP.

xv) 1², 5², 7², 73, ...

First term a₁ = 1²

Common difference, d = a₂ - a₁ = 25 - 1 = 24

Common difference, d = a₃ - a₂ = 49 - 25 = 24

Since a₂ - a₁ = a₃ - a₂, they form an AP

- Fifth term = a₁ + 4d = 1 + 4 × 24 = 1 + 96 = 97
- Sixth term = a₁ + 5d = 1 + 5 × 24 = 1 + 120 = 121
- Seventh term = a₁ + 6d = 1 + 6 × 24 = 1 + 144 = 145

1², 5², 7², 73 forms an AP with a common difference of 24. The next three terms are 97, 121, and 145.

**☛ Check: **NCERT Solutions for Class 10 Maths Chapter 5

**Video Solution:**

## Which of the following are APs? If they form an AP, find the common difference d and write three more termsi) 2, 4, 8, 16, ...ii) 2, 5/2 ,3, 7/2, ...iii) - 1.2, - 3.2, - 5.2, - 7.2, ...iv) - 10, - 6, - 2, 2, ...v) 3, 3 + √2, 3 + 2√2, 3 + 3√2, ...vi) 0.2, 0.22, 0.222, 0.2222, ...vii) 0, - 4, - 8, - 12, ...viii) - 1/2, - 1/2, - 1/2, - 1/2, ...ix) 1 ,3, 9, 27, ...x) a, 2a, 3a, 4a, ...xi) a, a², a³, a⁴, ...xii) √2, √8, √18, √32, ...xiii) √3, √6, √9, √12, ...xiv) 1², 3², 5², 7², ...xv) 1², 5², 7², 73, ...

Class 10 Maths NCERT Solutions Chapter 5 Exercise 5.1 Question 4

**Summary:**

i) 2, 4, 8, 16 are not in AP, because the common difference is not equal. ii) 2, 5/2 ,3, 7/2 forms an AP and the common difference is 1/2. The next three terms are 4, 9/2, 5 iii) - 1.2, - 3.2, - 5.2, - 7.2 forms an AP with common difference - 2. The next three terms of AP are - 9.2, - 11.2, - 13.2 iv) - 10, -6, - 2, 2 forms an AP with common difference 4 and next three terms are 6, 10, 14. v) 3, 3 + √2, 3 + 2√2, 3 + 3√2 forms an AP with common difference √2 and next three terms are 3 + 4√2, 3 + 5√2, 3 + 6√2 vi) 0.2, 0.22, 0.222, 0.2222 does not form an AP. vii) 0, - 4, - 8, - 12 forms an AP with a common difference of - 4. The next three terms are -16, -20, -24. viii) - 1/2, - 1/2, - 1/2, - 1/2 forms an AP with a common difference d = 0. The next three terms are - 1/2, - 1/2, - 1/2. ix) 1, 3, 9, 27 numbers do not form an AP. x) a, 2a, 3a, 4a forms an AP with a common difference d = a. The next three terms are 5a, 6a, 7a. xi) a, a^{2}, a^{3}, a^{4} does not form an AP. xii) √2, √8, √18, √32 forms an AP with a common difference of √2. The next three terms are √50, √72, √98. xiii) √3, √6, √9, √12 does not form an AP. xiv) 1², 3², 5², 7² does not form an AP. xv)1², 5², 7², 73 forms an AP with a common difference of 24. The next three terms are 97, 121, and 145.

**☛ Related Questions:**

- In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?(i) The taxi fare after each km when the fare is ₹ 15 for the first km and ₹ 8 for each additional km.(ii) The amount of air present in a cylinder when a vacuum pump removes 1/4 of the air remaining in the cylinder at a time.
- Write first four terms of AP, When the first term a and the common difference d are given as follows:i) a = 10, d = 10ii) a = - 2, d = 0iii) a = 4, d = - 3iv) a = - 1, d = 1/2v) a = - 1.25, d = - 0.25
- For the following APs, write the first term and the common difference:i) 3, 1, - 1, - 3....ii) - 5, - 1, 3,7....iii) 1/3, 5/3, 9/3, 13/3...iv) 0.6, 1.7, 2.8, 3.9....

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