Which of the following are A.P.'s ? If they form an A.P., find the common difference 'd' and write three more terms :
`(i) -10, -6, -2, 2, ..... " " (ii) 3, 3+sqrt(2), 3+2sqrt(2), 3+3sqrt(2), ...`
`(iii) 0, -4, -8, -12, .... " " (iv) a, 2a, 3a, 4a, .....`
(v)`sqrt(3), sqrt(6), sqrt(9), sqrt(12), .....`

Answer

To determine whether the given sequences are in Arithmetic Progression (A.P.), we need to check if the difference between consecutive terms is constant. If they are in A.P., we will also find the common difference 'd' and write three more terms of the sequence.

### Solution:

**(i)** Sequence: -10, -6, -2, 2

1. **Find the common difference (d):**
- d = -6 - (-10) = -6 + 10 = 4
- d = -2 - (-6) = -2 + 6 = 4
- d = 2 - (-2) = 2 + 2 = 4

Since the common difference is the same (4), this sequence is an A.P.

2. **Write three more terms:**
- Next term: 2 + 4 = 6
- Next term: 6 + 4 = 10
- Next term: 10 + 4 = 14

**Next three terms are:** 6, 10, 14

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**(ii)** Sequence: 3, 3 + √2, 3 + 2√2, 3 + 3√2

1. **Find the common difference (d):**
- d = (3 + √2) - 3 = √2
- d = (3 + 2√2) - (3 + √2) = 2√2 - √2 = √2
- d = (3 + 3√2) - (3 + 2√2) = 3√2 - 2√2 = √2

Since the common difference is the same (√2), this sequence is an A.P.

2. **Write three more terms:**
- Next term: (3 + 3√2) + √2 = 3 + 4√2
- Next term: (3 + 4√2) + √2 = 3 + 5√2
- Next term: (3 + 5√2) + √2 = 3 + 6√2

**Next three terms are:** 3 + 4√2, 3 + 5√2, 3 + 6√2

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**(iii)** Sequence: 0, -4, -8, -12

1. **Find the common difference (d):**
- d = -4 - 0 = -4
- d = -8 - (-4) = -8 + 4 = -4
- d = -12 - (-8) = -12 + 8 = -4

Since the common difference is the same (-4), this sequence is an A.P.

2. **Write three more terms:**
- Next term: -12 - 4 = -16
- Next term: -16 - 4 = -20
- Next term: -20 - 4 = -24

**Next three terms are:** -16, -20, -24

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**(iv)** Sequence: a, 2a, 3a, 4a

1. **Find the common difference (d):**
- d = 2a - a = a
- d = 3a - 2a = a
- d = 4a - 3a = a

Since the common difference is the same (a), this sequence is an A.P.

2. **Write three more terms:**
- Next term: 4a + a = 5a
- Next term: 5a + a = 6a
- Next term: 6a + a = 7a

**Next three terms are:** 5a, 6a, 7a

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**(v)** Sequence: √3, √6, √9, √12

1. **Find the common difference (d):**
- d = √6 - √3
- d = √9 - √6 = 3 - √6
- d = √12 - √9 = 2√3 - 3

Since the differences are not the same, this sequence is **not** an A.P.

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### Summary of Results:
1. (i) A.P. with d = 4, next terms: 6, 10, 14
2. (ii) A.P. with d = √2, next terms: 3 + 4√2, 3 + 5√2, 3 + 6√2
3. (iii) A.P. with d = -4, next terms: -16, -20, -24
4. (iv) A.P. with d = a, next terms: 5a, 6a, 7a
5. (v) Not an A.P.