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This section contains 2 paragraphs. Based upon one of paragraph 2 multiple choice questions and based on the other paragraphs 3 multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct
Paragraph for Question Nos. 10 and 11
a, b, c, d is an A.P. with integer terms and d = a2 + b2 + c2
The common difference is
- a)1
- b)2
- c)3
- d)none
Correct answer is option 'A'. Can you explain this answer?
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This section contains 2 paragraphs. Based upon one of paragraph 2 mult...
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This section contains 2 paragraphs. Based upon one of paragraph 2 mult...
Explanation:
Given Information:
- The terms a, b, c are in arithmetic progression (A.P.)
- The common difference is unknown
- The sum of the squares of the terms is d = a^2 + b^2 + c^2
Using the A.P. formula:
- For an A.P., the nth term can be represented as: a + (n-1)d
- The sum of the first n terms of an A.P. can be represented as: n/2 [2a + (n-1)d]
Sum of squares formula:
- d = a^2 + b^2 + c^2
Substitute the A.P. terms into the sum of squares formula:
- d = (a + (n-1)d)^2 + (a + 2(n-1)d)^2 + (a + 3(n-1)d)^2
Expand and simplify the equation:
- d = a^2 + 2ad(n-1) + d^2 + a^2 + 4ad(n-1) + 4d^2 + a^2 + 6ad(n-1) + 9d^2
- d = 3a^2 + 12ad(n-1) + 14d^2
Comparing with the given information:
- Since d = a^2 + b^2 + c^2, we can equate the coefficients of a, d, and constant terms
- 3a^2 = a^2, 12ad(n-1) = 0, 14d^2 = b^2 + c^2
From the second equation:
- 12ad(n-1) = 0
- This implies that the common difference, d, is 0
Therefore, the correct answer is option 'A' - 0.
Given Information:
- The terms a, b, c are in arithmetic progression (A.P.)
- The common difference is unknown
- The sum of the squares of the terms is d = a^2 + b^2 + c^2
Using the A.P. formula:
- For an A.P., the nth term can be represented as: a + (n-1)d
- The sum of the first n terms of an A.P. can be represented as: n/2 [2a + (n-1)d]
Sum of squares formula:
- d = a^2 + b^2 + c^2
Substitute the A.P. terms into the sum of squares formula:
- d = (a + (n-1)d)^2 + (a + 2(n-1)d)^2 + (a + 3(n-1)d)^2
Expand and simplify the equation:
- d = a^2 + 2ad(n-1) + d^2 + a^2 + 4ad(n-1) + 4d^2 + a^2 + 6ad(n-1) + 9d^2
- d = 3a^2 + 12ad(n-1) + 14d^2
Comparing with the given information:
- Since d = a^2 + b^2 + c^2, we can equate the coefficients of a, d, and constant terms
- 3a^2 = a^2, 12ad(n-1) = 0, 14d^2 = b^2 + c^2
From the second equation:
- 12ad(n-1) = 0
- This implies that the common difference, d, is 0
Therefore, the correct answer is option 'A' - 0.
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This section contains 2 paragraphs. Based upon one of paragraph 2 multiple choice questions and based on the other paragraphs 3 multiple choice questions have to be answered. Each of these questions has four choices (A), (B), (C) and (D) out of which ONLY ONE is correctParagraph for Question Nos. 10 and 11a, b, c, d is an A.P. with integer terms and d = a2 + b2 + c2The common difference isa)1b)2c)3d)noneCorrect answer is option 'A'. Can you explain this answer?
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