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The coefficients of three consecutive terms of (1 + x)n+5 are in the ratio 5 : 10 : 14. Then n = _______
Correct answer is '6'. Can you explain this answer?
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Given that the coefficients of three consecutive terms of (1 + x)n+5 are in the ratio 5 : 10 : 14.
Let the three consecutive terms be (n + 5)Ck, (n + 5)C(k + 1), and (n + 5)C(k + 2).
Then, we have (n + 5)Ck : (n + 5)C(k + 1) : (n + 5)C(k + 2) = 5 : 10 : 14.
Simplifying this expression, we get:
(n + 5)Ck / (n + 5)C(k + 1) = 1/2 and (n + 5)C(k + 1) / (n + 5)C(k + 2) = 7/5
Using the formula for binomial coefficients, we have:
(n + 5)! / k!(n + 5 - k)! / (n + 6)! / (k + 1)!(n + 4 - k)!
= 1/2 and (n + 6)! / (k + 1)!(n + 5 - k)! / (n + 7)! / (k + 2)!(n + 3 - k)!
= 7/5
Simplifying these expressions and cross-multiplying, we get:
n + 5 - k = 2k + 2 and n + 5 - (k + 1) = 5(k + 2) - (n + 6)
Simplifying further, we get:
n - k = 2 and n - 5k = -17
Solving these equations, we get:
k = 9 and n = 6
Therefore, the value of n is 6.
Given that the coefficients of three consecutive terms of (1 + x)n+5 are in the ratio 5 : 10 : 14.
Let the three consecutive terms be (n + 5)Ck, (n + 5)C(k + 1), and (n + 5)C(k + 2).
Then, we have (n + 5)Ck : (n + 5)C(k + 1) : (n + 5)C(k + 2) = 5 : 10 : 14.
Simplifying this expression, we get:
(n + 5)Ck / (n + 5)C(k + 1) = 1/2 and (n + 5)C(k + 1) / (n + 5)C(k + 2) = 7/5
Using the formula for binomial coefficients, we have:
(n + 5)! / k!(n + 5 - k)! / (n + 6)! / (k + 1)!(n + 4 - k)!
= 1/2 and (n + 6)! / (k + 1)!(n + 5 - k)! / (n + 7)! / (k + 2)!(n + 3 - k)!
= 7/5
Simplifying these expressions and cross-multiplying, we get:
n + 5 - k = 2k + 2 and n + 5 - (k + 1) = 5(k + 2) - (n + 6)
Simplifying further, we get:
n - k = 2 and n - 5k = -17
Solving these equations, we get:
k = 9 and n = 6
Therefore, the value of n is 6.
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The coefficients of three consecutive terms of(1 + x)n+5 are in the ratio 5 : 10 : 14. Then n = _______Correct answer is '6'. Can you explain this answer?
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The coefficients of three consecutive terms of(1 + x)n+5 are in the ratio 5 : 10 : 14. Then n = _______Correct answer is '6'. 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 The coefficients of three consecutive terms of(1 + x)n+5 are in the ratio 5 : 10 : 14. Then n = _______Correct answer is '6'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The coefficients of three consecutive terms of(1 + x)n+5 are in the ratio 5 : 10 : 14. Then n = _______Correct answer is '6'. Can you explain this answer?.
The coefficients of three consecutive terms of(1 + x)n+5 are in the ratio 5 : 10 : 14. Then n = _______Correct answer is '6'. 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 The coefficients of three consecutive terms of(1 + x)n+5 are in the ratio 5 : 10 : 14. Then n = _______Correct answer is '6'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The coefficients of three consecutive terms of(1 + x)n+5 are in the ratio 5 : 10 : 14. Then n = _______Correct answer is '6'. Can you explain this answer?.
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