JEE Exam > JEE Questions > The coefficients of three successive terms in...
Start Learning for Free
The coefficients of three successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will be
- a)11
- b)10
- c)12
- d)8
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
The coefficients of three successive terms in the expansion of (1 + x)...
Let coefficients of three consecutive terms i.e., (r + 1)th, (r + 2)th and (r + 3)th in expansion of (1 + x)n are 165.330 and 462 respectively then
coefficient of (r + 1)th term = nCr = 165
coefficient of (r + 2)th term = nCr + 1 = 330
coefficient of (r + 3)th term = nCr + 2 = 462
∴ n C r + 1 n C r = n - r r + 1 = 2
or, n - r = 2(r + 1)
or, r = 1 3 (n - 2)
and n C r + 2 n C r + 1 = n - r - 1 r + 2 = 231 165
or, 165(n - r - 1) = 231 (r + 2)
or, 165n - 627 = 396r
or, 165n - 627 = 396 x 1 3 (n - 2)
or, 165n - 627 = 132 (n - 2)
or, 33n = 363
∴ n = 11
coefficient of (r + 1)th term = nCr = 165
coefficient of (r + 2)th term = nCr + 1 = 330
coefficient of (r + 3)th term = nCr + 2 = 462
∴ n C r + 1 n C r = n - r r + 1 = 2
or, n - r = 2(r + 1)
or, r = 1 3 (n - 2)
and n C r + 2 n C r + 1 = n - r - 1 r + 2 = 231 165
or, 165(n - r - 1) = 231 (r + 2)
or, 165n - 627 = 396r
or, 165n - 627 = 396 x 1 3 (n - 2)
or, 165n - 627 = 132 (n - 2)
or, 33n = 363
∴ n = 11
Most Upvoted Answer
The coefficients of three successive terms in the expansion of (1 + x)...
Understanding the Problem
We are given coefficients of three successive terms in the binomial expansion of (1 + x)^n: 165, 330, and 462. We need to find the value of n.
Coefficients in Binomial Expansion
The coefficients in the expansion can be represented as:
- C(n, r) = n! / (r!(n-r)!)
Where C(n, r) is the coefficient of x^r in the expansion.
Setting Up the Equations
Let:
- C(n, r) = 165
- C(n, r+1) = 330
- C(n, r+2) = 462
Using the property of binomial coefficients, we can derive:
1. C(n, r+1) = (n - r) / (r + 1) * C(n, r)
2. C(n, r+2) = (n - r - 1) / (r + 2) * C(n, r + 1)
Formulating the Ratios
From the coefficients:
1. For C(n, r) and C(n, r+1):
- 330 = (n - r) / (r + 1) * 165
- This simplifies to: (n - r) / (r + 1) = 2
2. For C(n, r+1) and C(n, r+2):
- 462 = (n - r - 1) / (r + 2) * 330
- This simplifies to: (n - r - 1) / (r + 2) = 1.4
Solving the Equations
From the first equation:
- n - r = 2(r + 1) => n - r = 2r + 2 => n = 3r + 2
From the second equation:
- n - r - 1 = 1.4(r + 2) => n - r - 1 = 1.4r + 2.8 => n - r = 1.4r + 3.8 => n = 2.4r + 3.8
Equating the Two Expressions
Setting the two equations for n equal:
- 3r + 2 = 2.4r + 3.8
Solving gives:
- 0.6r = 1.8 => r = 3
Now substituting r back into n:
- n = 3(3) + 2 = 11.
Conclusion
Thus, the value of n is 11, confirming that the correct answer is option 'A'.
We are given coefficients of three successive terms in the binomial expansion of (1 + x)^n: 165, 330, and 462. We need to find the value of n.
Coefficients in Binomial Expansion
The coefficients in the expansion can be represented as:
- C(n, r) = n! / (r!(n-r)!)
Where C(n, r) is the coefficient of x^r in the expansion.
Setting Up the Equations
Let:
- C(n, r) = 165
- C(n, r+1) = 330
- C(n, r+2) = 462
Using the property of binomial coefficients, we can derive:
1. C(n, r+1) = (n - r) / (r + 1) * C(n, r)
2. C(n, r+2) = (n - r - 1) / (r + 2) * C(n, r + 1)
Formulating the Ratios
From the coefficients:
1. For C(n, r) and C(n, r+1):
- 330 = (n - r) / (r + 1) * 165
- This simplifies to: (n - r) / (r + 1) = 2
2. For C(n, r+1) and C(n, r+2):
- 462 = (n - r - 1) / (r + 2) * 330
- This simplifies to: (n - r - 1) / (r + 2) = 1.4
Solving the Equations
From the first equation:
- n - r = 2(r + 1) => n - r = 2r + 2 => n = 3r + 2
From the second equation:
- n - r - 1 = 1.4(r + 2) => n - r - 1 = 1.4r + 2.8 => n - r = 1.4r + 3.8 => n = 2.4r + 3.8
Equating the Two Expressions
Setting the two equations for n equal:
- 3r + 2 = 2.4r + 3.8
Solving gives:
- 0.6r = 1.8 => r = 3
Now substituting r back into n:
- n = 3(3) + 2 = 11.
Conclusion
Thus, the value of n is 11, confirming that the correct answer is option 'A'.
|
Explore Courses for JEE exam
|
|
Top Courses for JEEView all
The coefficients of three successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will bea)11b)10c)12d)8Correct answer is option 'A'. Can you explain this answer?
Question Description
The coefficients of three successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will bea)11b)10c)12d)8Correct answer is option 'A'. 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 successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will bea)11b)10c)12d)8Correct answer is option 'A'. 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 successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will bea)11b)10c)12d)8Correct answer is option 'A'. Can you explain this answer?.
The coefficients of three successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will bea)11b)10c)12d)8Correct answer is option 'A'. 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 successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will bea)11b)10c)12d)8Correct answer is option 'A'. 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 successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will bea)11b)10c)12d)8Correct answer is option 'A'. Can you explain this answer?.
Solutions for The coefficients of three successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will bea)11b)10c)12d)8Correct answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of The coefficients of three successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will bea)11b)10c)12d)8Correct answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The coefficients of three successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will bea)11b)10c)12d)8Correct answer is option 'A'. Can you explain this answer?, a detailed solution for The coefficients of three successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will bea)11b)10c)12d)8Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of The coefficients of three successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will bea)11b)10c)12d)8Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The coefficients of three successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will bea)11b)10c)12d)8Correct answer is option 'A'. Can you explain this answer? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup