If an=∑r=0^n (1)/(nrr), then ∑r=0^n (r)/(nr) equals(a) (n-1) an (b) n an (c) (1)/(2) n an (d) No

Name: Problem 31, Chapter 8: 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Physics
Uploaded: 2021-11-18T14:41:51-08:00
Duration: 1 min 6 s
Channel: Aman Gupta
Description: If an=∑r=0^n (1)/(nrr), then ∑r=0^n (r)/(nr) equals(a) (n-1) an (b) n an (c) (1)/(2) n an (d) None of the above

step 1 in this problem let b b equal to summation r equals to 0 to n r divided by ncr this can be writt

Question

If $a_{n}=\sum_{r=0}^{n} \frac{1}{n_{r}_{r}}$, then $\sum_{r=0}^{n} \frac{r}{n_{r}}$ equals(a) $(n-1) a_{n}$ (b) $n a_{n}$ (c) $\frac{1}{2} n a_{n}$ (d) None of the above

   If $a_{n}=\sum_{r=0}^{n} \frac{1}{n_{r}_{r}}$, then $\sum_{r=0}^{n} \frac{r}{n_{r}}$ equals(a) $(n-1) a_{n}$
(b) $n a_{n}$
(c) $\frac{1}{2} n a_{n}$
(d) None of the above

43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Physics

Disha Experts 16th Edition

Chapter 8, Problem 31

Instant Answer

Solved by Expert Aman Gupta

11/18/2021

Step 1

Step 1: First, let's rewrite the given expression $\sum_{r=0}^{n} \frac{r}{n_{r}}$ as $\sum_{r=0}^{n} \frac{r}{\binom{n}{r}}$ for clarity. Show more…

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If $a_{n}=\sum_{r=0}^{n} \frac{1}{n_{r}_{r}}$, then $\sum_{r=0}^{n} \frac{r}{n_{r}}$ equals(a) $(n-1) a_{n}$ (b) $n a_{n}$ (c) $\frac{1}{2} n a_{n}$ (d) None of the above

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Key Concepts

Symmetry in Weighted Sums

Leveraging the symmetry property of binomial coefficients is a powerful technique in summing series where each term is multiplied by a weight such as r. By matching terms corresponding to r and n - r, one can simplify the summation and relate it to an unweighted sum, often leading to elegant formulas.

Weighted Summation Techniques

Weighted sums are series where each term carries an additional factor (like r in this problem). Techniques that identify relationships between weighted and unweighted sums, especially through exploiting symmetry, are essential for deriving closed-form expressions and understanding deeper combinatorial identities.

Binomial Coefficients

Binomial coefficients, typically denoted as C(n, r), are fundamental in combinatorics and appear in the binomial theorem. Their properties, including symmetry (C(n, r) = C(n, n-r)), are crucial when working with series or sums involving these coefficients.

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