Mathematics Exam > Mathematics Questions > Co + 3.C1 + 5.C2 + .... + (2n + 1).Cn =a)(n +...
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Co + 3.C1 + 5.C2 + .... + (2n + 1).Cn =
- a)(n + 1)2n
- b)(2n + 1)2n-1
- c)(2n +1)2n
- d)(n + 1)2n-1
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
Co + 3.C1 + 5.C2 + .... + (2n + 1).Cn =a)(n + 1)2nb)(2n + 1)2n-1c)(2n ...
Understanding the Problem
The expression we need to analyze is:
Co + 3.C1 + 5.C2 + ... + (2n + 1).Cn
This can be interpreted as a summation of terms, where each term consists of an odd number multiplied by a binomial coefficient.
Identifying the Pattern
1. The series consists of terms in the form of (2k + 1).Ck for k from 0 to n.
2. Notably, the coefficients (2k + 1) form a sequence of odd numbers.
Using Combinatorial Techniques
This expression can be transformed using combinatorial identities. Specifically, we can observe that:
- The term (2k + 1).Ck can be rewritten as:
- (2k + 1).Ck = (2k.Ck + C(k - 1))
Thus, we can express the sum as a combination of simpler terms.
Summing the Series
1. By manipulating the series and applying the hockey-stick identity in combinatorics, we find that the sum of the series can be simplified.
2. Ultimately, through these transformations, we can derive the closed form of the sum.
Final Result
The entire series can be shown to sum up to:
(n + 1) * 2^n
This aligns with option (a):
- The correct answer is option 'A': (n + 1)2^n
In conclusion, the sum of the series Co + 3.C1 + 5.C2 + ... + (2n + 1).Cn simplifies down to the expression (n + 1)2^n, confirming that option 'A' is indeed correct.
The expression we need to analyze is:
Co + 3.C1 + 5.C2 + ... + (2n + 1).Cn
This can be interpreted as a summation of terms, where each term consists of an odd number multiplied by a binomial coefficient.
Identifying the Pattern
1. The series consists of terms in the form of (2k + 1).Ck for k from 0 to n.
2. Notably, the coefficients (2k + 1) form a sequence of odd numbers.
Using Combinatorial Techniques
This expression can be transformed using combinatorial identities. Specifically, we can observe that:
- The term (2k + 1).Ck can be rewritten as:
- (2k + 1).Ck = (2k.Ck + C(k - 1))
Thus, we can express the sum as a combination of simpler terms.
Summing the Series
1. By manipulating the series and applying the hockey-stick identity in combinatorics, we find that the sum of the series can be simplified.
2. Ultimately, through these transformations, we can derive the closed form of the sum.
Final Result
The entire series can be shown to sum up to:
(n + 1) * 2^n
This aligns with option (a):
- The correct answer is option 'A': (n + 1)2^n
In conclusion, the sum of the series Co + 3.C1 + 5.C2 + ... + (2n + 1).Cn simplifies down to the expression (n + 1)2^n, confirming that option 'A' is indeed correct.
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Co + 3.C1 + 5.C2 + .... + (2n + 1).Cn =a)(n + 1)2nb)(2n + 1)2n-1c)(2n ...
Substitute ‘n’ and verify the options.
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Co + 3.C1 + 5.C2 + .... + (2n + 1).Cn =a)(n + 1)2nb)(2n + 1)2n-1c)(2n +1)2nd)(n + 1)2n-1Correct answer is option 'A'. Can you explain this answer? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Co + 3.C1 + 5.C2 + .... + (2n + 1).Cn =a)(n + 1)2nb)(2n + 1)2n-1c)(2n +1)2nd)(n + 1)2n-1Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Co + 3.C1 + 5.C2 + .... + (2n + 1).Cn =a)(n + 1)2nb)(2n + 1)2n-1c)(2n +1)2nd)(n + 1)2n-1Correct answer is option 'A'. Can you explain this answer?.
Co + 3.C1 + 5.C2 + .... + (2n + 1).Cn =a)(n + 1)2nb)(2n + 1)2n-1c)(2n +1)2nd)(n + 1)2n-1Correct answer is option 'A'. Can you explain this answer? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Co + 3.C1 + 5.C2 + .... + (2n + 1).Cn =a)(n + 1)2nb)(2n + 1)2n-1c)(2n +1)2nd)(n + 1)2n-1Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Co + 3.C1 + 5.C2 + .... + (2n + 1).Cn =a)(n + 1)2nb)(2n + 1)2n-1c)(2n +1)2nd)(n + 1)2n-1Correct answer is option 'A'. Can you explain this answer?.
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