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If the pth, qth and rth terms of a AP be a, b, c respectively then show that a(q-r) b(r-p) c(p-q) =0?
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If the pth, qth and rth terms of a AP be a, b, c respectively then sho...
Let the first term of the given AP be "A" and let it's common difference be "d".
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If the pth, qth and rth terms of a AP be a, b, c respectively then sho...
Explanation:
We are given that the pth, qth and rth terms of an AP are a, b and c respectively. We need to prove that a(q-r) b(r-p) c(p-q) = 0.
Step 1: Find the common difference of the AP
Let d be the common difference of the AP. Then we have:
b = a + d(q - p) ....(1)
c = a + d(r - p) ....(2)
Step 2: Simplify the expression to be proved
a(q-r) b(r-p) c(p-q) = a(q-r) [a + d(r-p)] [a + d(q-p)] (p - q)
= a(q-r) [a^2 + ad(q-p) + ad(r-p) + d^2(q-p)(r-p)] (p - q)
Step 3: Use equations (1) and (2) to simplify the expression
a(q-r) b(r-p) c(p-q) = a(q-r) [a^2 + a(b-c) + d(q-p)(r-p)] (p - q)
= a(q-r) [a^2 - ad(p-q) + d(q-p)(r-p)] (p - q)
= a(q-r) [(a - d(p-r))^2 - d^2(p-q)(q-r)] (p - q)
Step 4: Use the identity a^2 - b^2 = (a + b)(a - b)
a(q-r) b(r-p) c(p-q) = a(q-r) [(a + d(p-r) + d(p-q) + d(q-r))(a - d(p-r) + d(p-q) - d(q-r))] (p - q)
= (a^2 - b^2)(c - a)(b - c)
= 0
Step 5: Conclusion
Hence, we have proved that a(q-r) b(r-p) c(p-q) = 0, which means that at least one of the factors a(q-r), b(r-p), c(p-q) is equal to zero.
We are given that the pth, qth and rth terms of an AP are a, b and c respectively. We need to prove that a(q-r) b(r-p) c(p-q) = 0.
Step 1: Find the common difference of the AP
Let d be the common difference of the AP. Then we have:
b = a + d(q - p) ....(1)
c = a + d(r - p) ....(2)
Step 2: Simplify the expression to be proved
a(q-r) b(r-p) c(p-q) = a(q-r) [a + d(r-p)] [a + d(q-p)] (p - q)
= a(q-r) [a^2 + ad(q-p) + ad(r-p) + d^2(q-p)(r-p)] (p - q)
Step 3: Use equations (1) and (2) to simplify the expression
a(q-r) b(r-p) c(p-q) = a(q-r) [a^2 + a(b-c) + d(q-p)(r-p)] (p - q)
= a(q-r) [a^2 - ad(p-q) + d(q-p)(r-p)] (p - q)
= a(q-r) [(a - d(p-r))^2 - d^2(p-q)(q-r)] (p - q)
Step 4: Use the identity a^2 - b^2 = (a + b)(a - b)
a(q-r) b(r-p) c(p-q) = a(q-r) [(a + d(p-r) + d(p-q) + d(q-r))(a - d(p-r) + d(p-q) - d(q-r))] (p - q)
= (a^2 - b^2)(c - a)(b - c)
= 0
Step 5: Conclusion
Hence, we have proved that a(q-r) b(r-p) c(p-q) = 0, which means that at least one of the factors a(q-r), b(r-p), c(p-q) is equal to zero.
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If the pth, qth and rth terms of a AP be a, b, c respectively then show that a(q-r) b(r-p) c(p-q) =0?
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If the pth, qth and rth terms of a AP be a, b, c respectively then show that a(q-r) b(r-p) c(p-q) =0? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If the pth, qth and rth terms of a AP be a, b, c respectively then show that a(q-r) b(r-p) c(p-q) =0? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the pth, qth and rth terms of a AP be a, b, c respectively then show that a(q-r) b(r-p) c(p-q) =0?.
If the pth, qth and rth terms of a AP be a, b, c respectively then show that a(q-r) b(r-p) c(p-q) =0? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If the pth, qth and rth terms of a AP be a, b, c respectively then show that a(q-r) b(r-p) c(p-q) =0? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the pth, qth and rth terms of a AP be a, b, c respectively then show that a(q-r) b(r-p) c(p-q) =0?.
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