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# Arithmetic progression

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### Arithmetic progression

1. 1. EX. NO. 2
2. 2. 1. Which of the following lists of numbers are Arithmetic Progressions? Justify. (i) 1, 3, 6, 10, ..... Let, t1 = 1, t2 = 3, t3 = 6, t4 = 10 t2 – t1 = 3 – 1 = 2 t3 – t2 = 6 – 3 = 3 t4 – t3 = 10 – 6 = 4 As, The difference between two consecutive terms is not constant. So, The sequence is not an A.P.
3. 3. ii. 3, 5, 7, 9, 11, ..... Sol. Let, t1 = 3, t2 = 5, t3 = 7, t4 = 9, t5 = 11 t2 – t1 = 5 – 3 = 2 t3 – t2 = 7 – 5 = 2 t4 – t3 = 9 – 7 = 2 t5 – t4 = 11 – 9 = 2 As, The difference between two consecutive terms is 2 which is constant. So, The sequence is an A.P.
4. 4. (iii) 1, 4, 7, 10, .... Sol. Let, t1 = 1, t2 = 4, t3 = 7, t4 = 10 t2 – t1 = 4 – 1 = 3 t3 – t2 = 7 – 4 = 3 t4 – t3 = 10 – 7 = 3 As, The difference between two consecutive terms 3 which is constant. So, The sequence is an A.P.
5. 5. (iv) 3, 6, 12, 24, .... Sol. t1 = 3, t2 = 6, t3 = 12, t4 = 24 t2 – t1 = 6 – 3 = 3 t3 – t2 = 12 – 6 = 6 t4 – t3 = 24 – 12 = 12 Since, The difference between two consecutive terms is not constant. So, The sequence is not an A.P.
6. 6. (v) 22, 26, 28, 31, ... Sol. t1 = 22, t2 = 26, t3 = 28, t4 = 31 t2 – t1 = 26 – 22 = 4 t3 – t2 = 28 – 26 = 2 t4 – t3 = 21 – 28 = 3 Since, The difference between two consecutive terms is not constant. So, The sequence is not an A.P.
7. 7. (vi) 0.5, 2, 3.5, 5, ... Sol. t1 = 0.5, t2 = 2, t3 = 3.5, t4 = 5 t2 – t1 = 2 – 0.5 = 1.5 t3 – t2 = 3.5 – 2 = 1.5 t4 – t3 = 5 – 3.5 = 1.5 As, The difference between two consecutive terms is 1.5 which is constant. So, The sequence is an A.P.
8. 8. (vii) 4, 3, 2, 1, .... Sol. t1 = 4, t2 = 3, t3 = 2, t4 = 1, t2 – t1 = 3 – 4 = – 1 t3 – t2 = 2 – 3 = – 1 t4 – t3 = 1 – 2 = – 1 As, The difference between two consecutive terms is –1 which is constant. So, The sequence is an A.P.
9. 9. (viii) – 10, – 13, – 16, – 19, ..... Sol. t1 = – 10, t2 = – 13, t3 = – 16, t4 = – 19 t2 – t1 = – 13 – (– 10) = – 13 + 10 = – 3 t3 – t2 = – 16 – (– 13) = – 16 + 13 = – 3 t4 – t3 = – 19 – (– 16) = – 19 + 16 = – 3 As, The difference between two consecutive terms is – 3 which is constant. So, The sequence is an A.P.
10. 10. 2. Write the first five terms of the following Arithmetic Progressions where, the common difference ‘d’ and the first term ‘a’ are given : (i) a = 2, d = 2.5 Sol. a = 2, d = 2.5 Here, t1 = a = 2 t2 = t1 + d = 2 + 2.5 = 4.5 t3 = t2 + d = 4.5 + 2.5 = 7 t4 = t3 + d = 7 + 2.5 = 9.5 t5 = t4 + d = 9.5 + 2.5 = 12 Thus, The first five terms of the A.P. are 2, 4.5, 7, 9.5 and 12.
11. 11. (ii) a = 10, d = – 3 Sol. a = 10, d = – 3 Here, t1 = a = 10 t2 = t1 + d = 10 + (– 3) = 10 – 3 = 7 t3 = t2 + d = 7 + (– 3) = 7 – 3 = 4 t4 = t3 + d = 4 + (– 3) = 4 – 3 = 1 t5 = t4 + d = 1 + (– 3) = 1 – 3 = – 2 Thus, The first five terms of the A.P. are 10, 7, 4, 1 and – 2.
12. 12. (iii) a = 4, d = 0 Sol. a = 4, d = 0 Here, t1 = a = 4 t2 = t1 + d = 4 + 0 = 4 t3 = t2 + d = 4 + 0 = 4 t4 = t3 + d = 4 + 0 = 4 t5 = t4 + d = 4 + 0 = 4 Thus, The first five terms of the A.P. are 4, 4, 4, 4 and 4.
13. 13. (iv) a = 5, d = 2 Sol. a = 5, d = 2 Here, t1 = a = 5 t2 = t1 + d = 5 + 2 = 7 t3 = t2 + d = 7 + 2 = 9 t4 = t3 + d = 9 + 2 = 11 t5 = t4 + d = 11 + 2 = 13 Thus, The first five terms of the A.P. are 5, 7, 9, 11 and 13.
14. 14. (v) a = 3, d = 4 Sol. a = 3, d = 4 Here, t1 = a = 3 t2 = t1 + d = 3 + 4 = 7 t3 = t2 + d = 7 + 4 = 11 t4 = t3 + d = 11 + 4 = 15 t5 = t4 + d = 15 + 4 = 19 Thus, The first five terms of the A.P. are 3, 7, 11, 15 and 19.
15. 15. (vi) a = 6, d = 6 Sol. a = 6, d = 6 Here, t1 = a = 6 t2 = t1 + d = 6 + 6 = 12 t3 = t2 + d = 12 + 6 = 18 t4 = t3 + d = 18 + 6 = 24 t5 = t4 + d = 24 + 6 = 30 Thus,The first five terms of A.P. are 6, 12, 18, 24 and 30.
The content structure of the provided text is a series of examples and solutions related to identifying and generating terms of arithmetic progressions (APs). Each example presents a sequence of numbers and asks whether it is an arithmetic progression, followed by a solution that calculates the differences between consecutive terms to justify the answer. Additionally, there are examples where the first term and common difference are given, and the task is to write the first five terms of the AP.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
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# Arithmetic progression

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### Arithmetic progression

1. 1. EX. NO. 2
2. 2. 1. Which of the following lists of numbers are Arithmetic Progressions? Justify. (i) 1, 3, 6, 10, ..... Let, t1 = 1, t2 = 3, t3 = 6, t4 = 10 t2 – t1 = 3 – 1 = 2 t3 – t2 = 6 – 3 = 3 t4 – t3 = 10 – 6 = 4 As, The difference between two consecutive terms is not constant. So, The sequence is not an A.P.
3. 3. ii. 3, 5, 7, 9, 11, ..... Sol. Let, t1 = 3, t2 = 5, t3 = 7, t4 = 9, t5 = 11 t2 – t1 = 5 – 3 = 2 t3 – t2 = 7 – 5 = 2 t4 – t3 = 9 – 7 = 2 t5 – t4 = 11 – 9 = 2 As, The difference between two consecutive terms is 2 which is constant. So, The sequence is an A.P.
4. 4. (iii) 1, 4, 7, 10, .... Sol. Let, t1 = 1, t2 = 4, t3 = 7, t4 = 10 t2 – t1 = 4 – 1 = 3 t3 – t2 = 7 – 4 = 3 t4 – t3 = 10 – 7 = 3 As, The difference between two consecutive terms 3 which is constant. So, The sequence is an A.P.
5. 5. (iv) 3, 6, 12, 24, .... Sol. t1 = 3, t2 = 6, t3 = 12, t4 = 24 t2 – t1 = 6 – 3 = 3 t3 – t2 = 12 – 6 = 6 t4 – t3 = 24 – 12 = 12 Since, The difference between two consecutive terms is not constant. So, The sequence is not an A.P.
6. 6. (v) 22, 26, 28, 31, ... Sol. t1 = 22, t2 = 26, t3 = 28, t4 = 31 t2 – t1 = 26 – 22 = 4 t3 – t2 = 28 – 26 = 2 t4 – t3 = 21 – 28 = 3 Since, The difference between two consecutive terms is not constant. So, The sequence is not an A.P.
7. 7. (vi) 0.5, 2, 3.5, 5, ... Sol. t1 = 0.5, t2 = 2, t3 = 3.5, t4 = 5 t2 – t1 = 2 – 0.5 = 1.5 t3 – t2 = 3.5 – 2 = 1.5 t4 – t3 = 5 – 3.5 = 1.5 As, The difference between two consecutive terms is 1.5 which is constant. So, The sequence is an A.P.
8. 8. (vii) 4, 3, 2, 1, .... Sol. t1 = 4, t2 = 3, t3 = 2, t4 = 1, t2 – t1 = 3 – 4 = – 1 t3 – t2 = 2 – 3 = – 1 t4 – t3 = 1 – 2 = – 1 As, The difference between two consecutive terms is –1 which is constant. So, The sequence is an A.P.
9. 9. (viii) – 10, – 13, – 16, – 19, ..... Sol. t1 = – 10, t2 = – 13, t3 = – 16, t4 = – 19 t2 – t1 = – 13 – (– 10) = – 13 + 10 = – 3 t3 – t2 = – 16 – (– 13) = – 16 + 13 = – 3 t4 – t3 = – 19 – (– 16) = – 19 + 16 = – 3 As, The difference between two consecutive terms is – 3 which is constant. So, The sequence is an A.P.
10. 10. 2. Write the first five terms of the following Arithmetic Progressions where, the common difference ‘d’ and the first term ‘a’ are given : (i) a = 2, d = 2.5 Sol. a = 2, d = 2.5 Here, t1 = a = 2 t2 = t1 + d = 2 + 2.5 = 4.5 t3 = t2 + d = 4.5 + 2.5 = 7 t4 = t3 + d = 7 + 2.5 = 9.5 t5 = t4 + d = 9.5 + 2.5 = 12 Thus, The first five terms of the A.P. are 2, 4.5, 7, 9.5 and 12.
11. 11. (ii) a = 10, d = – 3 Sol. a = 10, d = – 3 Here, t1 = a = 10 t2 = t1 + d = 10 + (– 3) = 10 – 3 = 7 t3 = t2 + d = 7 + (– 3) = 7 – 3 = 4 t4 = t3 + d = 4 + (– 3) = 4 – 3 = 1 t5 = t4 + d = 1 + (– 3) = 1 – 3 = – 2 Thus, The first five terms of the A.P. are 10, 7, 4, 1 and – 2.
12. 12. (iii) a = 4, d = 0 Sol. a = 4, d = 0 Here, t1 = a = 4 t2 = t1 + d = 4 + 0 = 4 t3 = t2 + d = 4 + 0 = 4 t4 = t3 + d = 4 + 0 = 4 t5 = t4 + d = 4 + 0 = 4 Thus, The first five terms of the A.P. are 4, 4, 4, 4 and 4.
13. 13. (iv) a = 5, d = 2 Sol. a = 5, d = 2 Here, t1 = a = 5 t2 = t1 + d = 5 + 2 = 7 t3 = t2 + d = 7 + 2 = 9 t4 = t3 + d = 9 + 2 = 11 t5 = t4 + d = 11 + 2 = 13 Thus, The first five terms of the A.P. are 5, 7, 9, 11 and 13.
14. 14. (v) a = 3, d = 4 Sol. a = 3, d = 4 Here, t1 = a = 3 t2 = t1 + d = 3 + 4 = 7 t3 = t2 + d = 7 + 4 = 11 t4 = t3 + d = 11 + 4 = 15 t5 = t4 + d = 15 + 4 = 19 Thus, The first five terms of the A.P. are 3, 7, 11, 15 and 19.
15. 15. (vi) a = 6, d = 6 Sol. a = 6, d = 6 Here, t1 = a = 6 t2 = t1 + d = 6 + 6 = 12 t3 = t2 + d = 12 + 6 = 18 t4 = t3 + d = 18 + 6 = 24 t5 = t4 + d = 24 + 6 = 30 Thus,The first five terms of A.P. are 6, 12, 18, 24 and 30.
user
The content structure of the provided text is a series of examples and solutions related to identifying and generating terms of arithmetic progressions (APs). Each example presents a sequence of numbers and asks whether it is an arithmetic progression, followed by a solution that calculates the differences between consecutive terms to justify the answer. Additionally, there are examples where the first term and common difference are given, and the task is to write the first five terms of the AP.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS