JEE Exam > JEE Questions > Find the largest co-efficient in the expansio...
Start Learning for Free
Find the largest co-efficient in the expansion of (1 + x)n, given that the sum of co-efficients of the terms in its expansion is 4096. [REE 2000 (Mains)]
Correct answer is '924'. Can you explain this answer?
Verified Answer
Find the largest co-efficient in the expansion of (1 + x)n, given that...
We know that, the coefficients in a binomial expansion is obtained by replacing each variable by unit in the given expression.
Therefore, sum of the coefficients in (a+b)^n
= 4096=(1+1)n
⇒ 4096=(2)n
⇒ (2)12=(2)n
⇒ n=12
Here n is even, so the greatest coefficient is nCn/2 i.e., 12C6 = 924
Therefore, sum of the coefficients in (a+b)^n
= 4096=(1+1)n
⇒ 4096=(2)n
⇒ (2)12=(2)n
⇒ n=12
Here n is even, so the greatest coefficient is nCn/2 i.e., 12C6 = 924
Most Upvoted Answer
Find the largest co-efficient in the expansion of (1 + x)n, given that...
Given:
The expansion of (1 + x)^n has a sum of coefficients equal to 4096.
To find:
The largest coefficient in the expansion of (1 + x)^n.
Solution:
Step 1: Expansion of (1 + x)^n
The expansion of (1 + x)^n can be written as:
(1 + x)^n = C(n, 0) + C(n, 1)x + C(n, 2)x^2 + ... + C(n, n)x^n
Where C(n, r) represents the binomial coefficient, which is given by the formula:
C(n, r) = n! / (r!(n-r)!)
Step 2: Sum of coefficients
The sum of coefficients in the expansion of (1 + x)^n is equal to the value of the expression when x is replaced by 1, i.e., x = 1. Therefore, substituting x = 1 in the expansion, we get:
(1 + 1)^n = C(n, 0) + C(n, 1) + C(n, 2) + ... + C(n, n)
Simplifying this expression gives us:
2^n = C(n, 0) + C(n, 1) + C(n, 2) + ... + C(n, n)
Given that the sum of coefficients is 4096, we can write the equation as:
2^n = 4096
Step 3: Solving for n
To find the value of n, we need to solve the equation 2^n = 4096. Taking the logarithm of both sides, we get:
log2(2^n) = log2(4096)
n log2(2) = log2(4096)
n = log2(4096) / log2(2)
n = 12
Therefore, the value of n is 12.
Step 4: Finding the largest coefficient
The largest coefficient in the expansion of (1 + x)^n occurs when x^n is the term with the largest power of x. In this case, the largest power of x is x^6, which occurs when r = 6 in the binomial coefficient C(n, r).
Therefore, the largest coefficient is C(12, 6) or 12C6.
Hence, the correct answer is 12C6.
The expansion of (1 + x)^n has a sum of coefficients equal to 4096.
To find:
The largest coefficient in the expansion of (1 + x)^n.
Solution:
Step 1: Expansion of (1 + x)^n
The expansion of (1 + x)^n can be written as:
(1 + x)^n = C(n, 0) + C(n, 1)x + C(n, 2)x^2 + ... + C(n, n)x^n
Where C(n, r) represents the binomial coefficient, which is given by the formula:
C(n, r) = n! / (r!(n-r)!)
Step 2: Sum of coefficients
The sum of coefficients in the expansion of (1 + x)^n is equal to the value of the expression when x is replaced by 1, i.e., x = 1. Therefore, substituting x = 1 in the expansion, we get:
(1 + 1)^n = C(n, 0) + C(n, 1) + C(n, 2) + ... + C(n, n)
Simplifying this expression gives us:
2^n = C(n, 0) + C(n, 1) + C(n, 2) + ... + C(n, n)
Given that the sum of coefficients is 4096, we can write the equation as:
2^n = 4096
Step 3: Solving for n
To find the value of n, we need to solve the equation 2^n = 4096. Taking the logarithm of both sides, we get:
log2(2^n) = log2(4096)
n log2(2) = log2(4096)
n = log2(4096) / log2(2)
n = 12
Therefore, the value of n is 12.
Step 4: Finding the largest coefficient
The largest coefficient in the expansion of (1 + x)^n occurs when x^n is the term with the largest power of x. In this case, the largest power of x is x^6, which occurs when r = 6 in the binomial coefficient C(n, r).
Therefore, the largest coefficient is C(12, 6) or 12C6.
Hence, the correct answer is 12C6.
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
Find the largest co-efficient in the expansion of (1 + x)n, given that the sum of co-efficients of the terms in its expansion is 4096. [REE 2000 (Mains)]Correct answer is '924'. Can you explain this answer?
Question Description
Find the largest co-efficient in the expansion of (1 + x)n, given that the sum of co-efficients of the terms in its expansion is 4096. [REE 2000 (Mains)]Correct answer is '924'. 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 Find the largest co-efficient in the expansion of (1 + x)n, given that the sum of co-efficients of the terms in its expansion is 4096. [REE 2000 (Mains)]Correct answer is '924'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the largest co-efficient in the expansion of (1 + x)n, given that the sum of co-efficients of the terms in its expansion is 4096. [REE 2000 (Mains)]Correct answer is '924'. Can you explain this answer?.
Find the largest co-efficient in the expansion of (1 + x)n, given that the sum of co-efficients of the terms in its expansion is 4096. [REE 2000 (Mains)]Correct answer is '924'. 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 Find the largest co-efficient in the expansion of (1 + x)n, given that the sum of co-efficients of the terms in its expansion is 4096. [REE 2000 (Mains)]Correct answer is '924'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the largest co-efficient in the expansion of (1 + x)n, given that the sum of co-efficients of the terms in its expansion is 4096. [REE 2000 (Mains)]Correct answer is '924'. Can you explain this answer?.
Solutions for Find the largest co-efficient in the expansion of (1 + x)n, given that the sum of co-efficients of the terms in its expansion is 4096. [REE 2000 (Mains)]Correct answer is '924'. 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 Find the largest co-efficient in the expansion of (1 + x)n, given that the sum of co-efficients of the terms in its expansion is 4096. [REE 2000 (Mains)]Correct answer is '924'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Find the largest co-efficient in the expansion of (1 + x)n, given that the sum of co-efficients of the terms in its expansion is 4096. [REE 2000 (Mains)]Correct answer is '924'. Can you explain this answer?, a detailed solution for Find the largest co-efficient in the expansion of (1 + x)n, given that the sum of co-efficients of the terms in its expansion is 4096. [REE 2000 (Mains)]Correct answer is '924'. Can you explain this answer? has been provided alongside types of Find the largest co-efficient in the expansion of (1 + x)n, given that the sum of co-efficients of the terms in its expansion is 4096. [REE 2000 (Mains)]Correct answer is '924'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Find the largest co-efficient in the expansion of (1 + x)n, given that the sum of co-efficients of the terms in its expansion is 4096. [REE 2000 (Mains)]Correct answer is '924'. 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