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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.
explanation: 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 forms an arithmetic progression. The solutions justify the answer by calculating the differences between consecutive terms and checking if they are constant. Additionally, there are examples where the first term and the common difference are given, and the task is to write the first five terms of the AP.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
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  1. Briefly describe the document and mention if it contains a question or a question and answer.
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distilabel_metadata.content: Upcoming SlideShare
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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.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: 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 forms an arithmetic progression. The solutions justify the answer by calculating the differences between consecutive terms and checking if they are constant. Additionally, there are examples where the first term and the common difference are given, and the task is to write the first five terms of the AP.
  
  Label: multiple-question-and-answer