# 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) - ½, - ½, - ½, - ½, ...

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

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

xi) a, a², a³, a^{4}, ...

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 the common difference.

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

First term a = 2

Common difference d = a_{2} - a_{1} = 4 - 2 = 2

Common difference d = a_{3} - a_{2} = 8 - 4 = 4

(a_{3} - a_{2}) ≠ (a_{2} - a_{1})

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 a = 2

Common difference d = a_{2} - a_{1} = 5/2 - 2 = (5 - 4)/2 = ½

Common difference d = a_{3} - a_{2} = 3 - 5/2 = (6 - 5)/2 = ½

Since a_{3} - a_{2} = a_{2} - a_{1}.

2, 5/2 ,3, 7/2 forms an AP and common difference is ½

The next three terms are:

- Fifth term = a + 4d

= 2 + 4 × 1/2

= 2 + 2

= 4

- Sixth term = a + 5d

= 2 + 5 × ½

= 2 + 5/2

= (4 + 5) / 2

= 9/2

- Seventh term = a + 6d

= 2 + 6 × ½

= 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 a = - 1.2

Common difference d = a_{2} - a_{1} = -3.2 - (-1.2) = -3.2 + 1.2 = - 2

Common difference d = a_{3} - a_{2} = -5.2 - (-3.2)

= - 5.2 + 3.2 = - 2

Since a_{3} - a_{2} = a_{2} - a_{1,} it forms an AP.

- Fifth term = a + 4d

= - 1.2 + 4(- 2)

= -1.2 - 8

= - 9.2

- Sixth term = a + 5d

= - 1.2 + 5(- 2)

= - 1.2 - 10

= - 11.2

The seventh term = a + 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 (a) = - 10

Common difference (d) is a_{2} - a_{1}

= - 6 - (- 10)

= - 6 + 10

Common difference (d) is = a_{3} - a_{2}

= - 2 - (- 6)

= - 2 + 6

= 4

Since a_{3} - a_{2} = a_{2} - a_{1}, - 10, - 6, - 2, 2, ... forms an AP

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

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

v) 3, 3 + √2, 3 + 2√2, 3 + 3√2, ...

First term (a) = 3

Common difference (d) is = a_{2} - a_{1}

= 3 + √2 - 3

= √2

Common difference (d) is = a_{3 }- a_{2}

= 3 + 2√2 - (3 + √2)

= 3 + 2√2 - 3 - √2

= √2

Since a_{3} - a_{2} = a_{2} - a_{1}, 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 = a + 4d

= 3 + 4 × √2

= 3 + 4√2

- Sixth term = a + 5d

= 3 + 5 × √2

= 3 + 5√2

- Seventh term = a + 6d

= 3 + 6 × √2

= 3 + 6√2

It is 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 = 0.2

Common difference d = a_{2 }- a_{1}

= 0.22 - 0.2

= 0.02

Common difference d = a_{3 }- a_{2}

= 0.222 - 0.220

= 0.002

Since (a_{3} - a_{2}) ≠ (a_{2} - a_{1}), 0.2, 0.22, 0.222, 0.2222, ... do not forms an AP.

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

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

First term (a) = 0

Common difference (d) is = a_{2} - a_{1} = - 4 - 0 = - 4

Common difference (d) is = a_{3} - a_{2} = -8 - (- 4) = - 8 + 4 = - 4

Since a_{3} - a_{2} = a_{2} - a_{1}, it forms an AP.

- Fifth term = a + 4d

= 0 + 4(- 4)

= - 16

- Sixth term-= a + 5d

= 0 + 5(- 4)

= - 20

- Seventh term = a + 6d

= 0 + 6(- 4)

= - 24

The given numbers form an AP with a difference of - 4. The next three terms are -16, -20, -24.

viii) - ½, - ½, - ½, - ½,....

First term = - ½

Common difference d = a_{2} - a_{1}

= - ½ - (- ½)

= - ½ + ½

= 0

Common difference d = a_{3} - a_{2}

= - ½ - (- ½)

= - ½ + ½

= 0

Since a_{3} - a_{2} = a_{2} - a_{1}, the list of numbers forms an AP.

- Fifth term = a + 4d

= - ½ + 4 (0)

= - ½

- Sixth term = a + 5d

= - ½ + 5 (0)

= - ½

- Seventh term = a + 6d

= - ½ + 6 (0)

= - ½

The given list of numbers forms an AP with a common difference d = 0. The next three terms are - ½, - ½, - ½,

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

First term (a) = 1

Common difference (d) = a_{2} - a_{1} = 3 - 1= 2

Common difference (d) = a_{3} - a_{2} = 9 - 3 = 6

Since a_{2} - a_{1} ≠ a_{3} - a_{2}, the given list of numbers does not form an AP.

The given list of numbers does not form an AP.

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

First term (a) = a

Common difference, d = a_{2} - a_{1}

= 2a - a

= a

Common difference, d = a_{3} - a_{2}

= 3a - 2a

= a

Since a_{3} - a_{2} = a_{2} - a_{1}, a, 2a, 3a, 4a, ... forms an AP.

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

The given list of numbers 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 (a) = a

Common difference, = a_{2} - a_{1}

= a^{2 }- a

= a (a - 1)

Common difference, d = a_{3} - a_{2}

= a^{3 }- a^{2}

= a^{2} (a - 1)

Since a_{2} - a_{1} ≠ a_{3} - a_{2}, the given list of numbers does not form an AP.

xii) √2, √8, √18, √32......

First term (a) = √2

Common difference, d = a_{2} - a_{1}

= √8 - √2

= 2√2 - √2

= √2

Common difference d = a_{3 }- a_{2}

= √18 - √8

= 3√2 - 2√2

= √2

Since a_{2} - a_{1} = a_{3} - a_{2}, the given numbers form an AP.

- Fifth term = a + 4d

= √2 + 4√2

= 5√2

= √25 × 2

= √50

- Sixth term = a + 5d

= √2 + 5√2

= 6√2

= √36 × 2

= √72

- Seventh term-= a + 6d

= √2 + 6√2

= 7√2

= √49 × 2

= √98

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

xiii) √3, √6, √9, √12, ...

First term (a) = √3

Common difference d = a_{2} - a_{1}

= √6 - √3

= √3 × 2 - √3

= √3 (√2 - 1)

Common difference d = a_{3} - a_{2} = √9 - √6

= √3 × 3 - √3 × 2

= √3 (√3 - √2)

Since a_{2} - a_{1} ≠ a_{3} - a_{2}, the given list of numbers does not form an AP.

xiv) 1², 3², 5², 7², ...

First tem (a) = 1²

Common difference, d = a_{2} - a_{1} = 9 - 1 = 8

Common difference, d = a_{3} - a_{2} = 25 - 9 = 16

Since a_{2} - a_{1} ≠ a_{3} - a_{2}, the given list of numbers does not form an AP.

xv) 1², 5², 7², 73, ...

First term (a) = 1²

Common difference, d = a_{2} - a_{1} = 25 - 1 = 24

Common difference, d = a_{3} - a_{2} = 49 - 25 = 24

Since a_{2} - a_{1} = a_{3} - a_{2}, they form an AP

- Fifth term = a + 4d

= 1 + 4 × 24

= 1 + 96

= 97

- Sixth term = a + 5d

= 1 + 5 × 24

= 1 + 120

= 121

- Seventh term = a + 6d

= 1 + 6 × 24

= 1 + 144

= 145

The list of numbers forms an AP with a common difference of 24. The next three terms are 97, 121, and 145.

**Video Solution:**

### Class 10 Maths NCERT Solutions - Chapter 5 Exercise 5.1 Question 4:

Final Answer

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 ½. 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) It is an AP with common difference √2 and the next three terms are 3 + 4√2, 3 + 5√2, 3 + 6√2

v) The given list of numbers does not form an AP.

vii) The given numbers form an AP with a difference of -4. The next three terms are -16, -20, -24

viii) The given list of numbers forms an AP with a common difference d = 0. The next three terms are - ½ , - ½ ,- ½

ix) The given list of numbers does not form an AP.

x) The given list of numbers forms an AP with a common difference d = a. The next three terms are 5a, 6a, 7a.

xi) The given list of numbers does not form an AP.

xii) The list of numbers forms an AP with a common difference √2. The next three terms are √50, √72, √98

xiii) The given list of numbers does not form an AP.

xiv) The given list of numbers does not form an AP.

xv)The list of numbers forms an AP with a common difference of 24. The next three terms are 97, 121, and 145.