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:
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.
- 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
- 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
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.
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.
- 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.
a, 2a, 3a, 4a forms an AP with a common difference d = a. The next three terms are 5a, 6a, 7a. xi) a, a2, a3, a4...... First term a₁ = a Common difference, = a₂ - a₁ = a2 - a = a (a - 1) Common difference, d = a₃ - a₂ = a3 - a2 = a2 (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.
√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
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, a2, a3, a4 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:
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