Question

if \(a_{n}={ }^{n} \sum_{r=0}\left[1 /\left({ }^{n} C_{r}\right)\right]\) then value of \(^{n} \sum_{r=0}\left[(n-2 r) /\left({ }^{n} C_{r}\right)\right]\) is (a) \((\mathrm{n} / 2) \mathrm{a}_{\mathrm{n}}\) (b) \((1 / 4) \mathrm{a}_{\mathrm{n}}\) (c) \(\mathrm{n} \cdot \mathrm{a}_{\mathrm{n}}\) (d) none of these.

Short answer

The correct answer is (d) None of these.

Step-by-step solution

Apply the Binomial Theorem

Recall the binomial theorem: \[(x+y)^n = \sum_{r=0}^n\binom{n}{r}x^{n-r}y^r\] Now, let \[x = 1\] and \[y = -1\], and we can simplify: \[(1-1)^n = \sum_{r=0}^n\binom{n}{r}(1)^{n-r}(-1)^r\] This gives us: \[0 = \sum_{r=0}^n\binom{n}{r}(n-2r)\] Step 1 shows us that our sum equals 0.

Compare with the given options

Now, let's compare this result with the multiple-choice options given.

Option (a)

\[(\frac{n}{2}) a_n = \frac{n}{2}\sum_{r=0}^n\frac{1}{\binom{n}{r}}\] This doesn't match our result from Step 1, so this is not the correct answer.

Option (b)

\[(\frac{1}{4})a_n = \frac{1}{4} \sum_{r=0}^n\frac{1}{\binom{n}{r}}\] This doesn't match our result from Step 1, so this is not the correct answer.

Option (c)

\[na_n = n \sum_{r=0}^n\frac{1}{\binom{n}{r}}\] This doesn't match our result from Step 1, so this is not the correct answer. Since none of the options (a), (b), or (c) match the result we obtained in Step 1, the correct answer must be:

Answer

(d) None of these.

Key concepts

JEE Advanced Mathematics

The Joint Entrance Examination (JEE) Advanced is one of the most challenging engineering entrance exams in India. It includes a section on Mathematics that tests a student's understanding of various mathematical concepts, including calculus, algebra, trigonometry, and geometry.

Within the Mathematics section, problems can range from straightforward to complex, involving deep conceptual knowledge and problem-solving skills. Concepts from JEE Advanced Mathematics often intertwine, requiring students to apply multiple topics to solve a single problem. It's an assessment that truly measures a student's mathematical abilities, preparing them for the rigorous math they will encounter in their future engineering courses.

Combinatorics

Combinatorics is a branch of mathematics dealing with the counting, arrangement, and combination of objects. It's foundational for understanding probability, statistics, and even computer science algorithms.

In the context of JEE Advanced Mathematics, combinatorics problems can include finding the number of ways to arrange objects, count subsets, or solve more complex optimization problems. This section of mathematics teaches students the art of breaking down large, complex problems into manageable parts, a skill that is useful across all areas of problem-solving.

Binomial Theorem

The Binomial Theorem is a critical mathematical principle that helps in the expansion of powers of a binomial expression. This theorem states that \[(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\] where \(\binom{n}{k}\) are the binomial coefficients, representing the number of ways to choose k elements out of n without considering the order.

Understanding this theorem is particularly important for JEE aspirants as it is widely applied in polynomial expansions, probability, and series. It also underpins many results in combinatorics. The theorem can simplistically be seen as a formula to understand the algebra of coefficients and is a gateway to exploring deeper relationships in algebra and number theory.

Summation Notation

Summation notation, also known as sigma notation, is a convenient way to express a long sum of terms in a compact form. It's denoted by the Greek letter sigma (\(\Sigma\)) and is particularly useful in dealing with series and sequences.

In JEE Mathematics, learning how to use summation notation is fundamental. It allows for concise expression of long sums and can be used in conjunction with other mathematical tools such as the Binomial Theorem to solve complex problems. This notation is also crucial in statistics for representing data sums and in calculus for defining integrals and series.