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If the coefficients of rth, (r + 1)th and (r + 2)th terms in the binomial expansion of (1 + y)m are in A.P., then m and r satisfy the equation-
[AIEEE-2005]
- a)m2 –m (4r – 1) + 4r2 – 2 = 0
- b)m2 – m (4r + 1) + 4r2 + 2 = 0
- c)m2 – m (4r + 1) + 4r2 – 2 = 0
- d)m2 – m (4r – 1) + 4r2 + 2 = 0
Correct answer is option 'C'. Can you explain this answer?
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Understanding the Problem
In the binomial expansion of (1 + y)^m, the coefficients of the rth, (r + 1)th, and (r + 2)th terms can be expressed using the binomial coefficient:
- Coefficient of rth term: C(m, r)
- Coefficient of (r + 1)th term: C(m, r + 1)
- Coefficient of (r + 2)th term: C(m, r + 2)
Condition for A.P.
The coefficients are in Arithmetic Progression (A.P.) if:
2 * C(m, r + 1) = C(m, r) + C(m, r + 2)
Using Binomial Coefficient Properties
Using the property of binomial coefficients:
- C(m, r + 1) = C(m, r) * (m - r) / (r + 1)
- C(m, r + 2) = C(m, r + 1) * (m - r - 1) / (r + 2)
Substituting these into the A.P. condition gives:
2 * C(m, r) * (m - r) / (r + 1) = C(m, r) + C(m, r) * (m - r) * (m - r - 1) / ((r + 1)(r + 2))
Simplifying the Equation
After simplification, we can derive a quadratic equation in terms of m:
m^2 - m(4r + 1) + 4r^2 - 2 = 0
This corresponds to option (C) from the given choices.
Conclusion
Thus, the relationship between m and r can be succinctly captured by the equation in option C, confirming the coefficients of the specified terms being in Arithmetic Progression.
In the binomial expansion of (1 + y)^m, the coefficients of the rth, (r + 1)th, and (r + 2)th terms can be expressed using the binomial coefficient:
- Coefficient of rth term: C(m, r)
- Coefficient of (r + 1)th term: C(m, r + 1)
- Coefficient of (r + 2)th term: C(m, r + 2)
Condition for A.P.
The coefficients are in Arithmetic Progression (A.P.) if:
2 * C(m, r + 1) = C(m, r) + C(m, r + 2)
Using Binomial Coefficient Properties
Using the property of binomial coefficients:
- C(m, r + 1) = C(m, r) * (m - r) / (r + 1)
- C(m, r + 2) = C(m, r + 1) * (m - r - 1) / (r + 2)
Substituting these into the A.P. condition gives:
2 * C(m, r) * (m - r) / (r + 1) = C(m, r) + C(m, r) * (m - r) * (m - r - 1) / ((r + 1)(r + 2))
Simplifying the Equation
After simplification, we can derive a quadratic equation in terms of m:
m^2 - m(4r + 1) + 4r^2 - 2 = 0
This corresponds to option (C) from the given choices.
Conclusion
Thus, the relationship between m and r can be succinctly captured by the equation in option C, confirming the coefficients of the specified terms being in Arithmetic Progression.
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If the coefficients of rth, (r + 1)th and (r + 2)th terms in the binomial expansion of (1 + y)m are in A.P., then m and r satisfy the equation-[AIEEE-2005]a)m2 –m (4r – 1) + 4r2 – 2 = 0b)m2 – m (4r + 1) + 4r2 + 2 = 0c)m2 – m (4r + 1) + 4r2 – 2 = 0d)m2 – m (4r – 1) + 4r2 + 2 = 0Correct answer is option 'C'. Can you explain this answer?
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If the coefficients of rth, (r + 1)th and (r + 2)th terms in the binomial expansion of (1 + y)m are in A.P., then m and r satisfy the equation-[AIEEE-2005]a)m2 –m (4r – 1) + 4r2 – 2 = 0b)m2 – m (4r + 1) + 4r2 + 2 = 0c)m2 – m (4r + 1) + 4r2 – 2 = 0d)m2 – m (4r – 1) + 4r2 + 2 = 0Correct answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the coefficients of rth, (r + 1)th and (r + 2)th terms in the binomial expansion of (1 + y)m are in A.P., then m and r satisfy the equation-[AIEEE-2005]a)m2 –m (4r – 1) + 4r2 – 2 = 0b)m2 – m (4r + 1) + 4r2 + 2 = 0c)m2 – m (4r + 1) + 4r2 – 2 = 0d)m2 – m (4r – 1) + 4r2 + 2 = 0Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the coefficients of rth, (r + 1)th and (r + 2)th terms in the binomial expansion of (1 + y)m are in A.P., then m and r satisfy the equation-[AIEEE-2005]a)m2 –m (4r – 1) + 4r2 – 2 = 0b)m2 – m (4r + 1) + 4r2 + 2 = 0c)m2 – m (4r + 1) + 4r2 – 2 = 0d)m2 – m (4r – 1) + 4r2 + 2 = 0Correct answer is option 'C'. Can you explain this answer?.
If the coefficients of rth, (r + 1)th and (r + 2)th terms in the binomial expansion of (1 + y)m are in A.P., then m and r satisfy the equation-[AIEEE-2005]a)m2 –m (4r – 1) + 4r2 – 2 = 0b)m2 – m (4r + 1) + 4r2 + 2 = 0c)m2 – m (4r + 1) + 4r2 – 2 = 0d)m2 – m (4r – 1) + 4r2 + 2 = 0Correct answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the coefficients of rth, (r + 1)th and (r + 2)th terms in the binomial expansion of (1 + y)m are in A.P., then m and r satisfy the equation-[AIEEE-2005]a)m2 –m (4r – 1) + 4r2 – 2 = 0b)m2 – m (4r + 1) + 4r2 + 2 = 0c)m2 – m (4r + 1) + 4r2 – 2 = 0d)m2 – m (4r – 1) + 4r2 + 2 = 0Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the coefficients of rth, (r + 1)th and (r + 2)th terms in the binomial expansion of (1 + y)m are in A.P., then m and r satisfy the equation-[AIEEE-2005]a)m2 –m (4r – 1) + 4r2 – 2 = 0b)m2 – m (4r + 1) + 4r2 + 2 = 0c)m2 – m (4r + 1) + 4r2 – 2 = 0d)m2 – m (4r – 1) + 4r2 + 2 = 0Correct answer is option 'C'. Can you explain this answer?.
Solutions for If the coefficients of rth, (r + 1)th and (r + 2)th terms in the binomial expansion of (1 + y)m are in A.P., then m and r satisfy the equation-[AIEEE-2005]a)m2 –m (4r – 1) + 4r2 – 2 = 0b)m2 – m (4r + 1) + 4r2 + 2 = 0c)m2 – m (4r + 1) + 4r2 – 2 = 0d)m2 – m (4r – 1) + 4r2 + 2 = 0Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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