If a n - r -0 n 1- n C r then - r -0 n r - r C T equalsA- nanB- None of theseC- n-1 anD- 1-2 na n

The correct option is D 1/2n a_n Let b = ∑_r=0^n r/^nC_r = ∑_r=0^n n (n r)/^nC_r = n ∑_r=0^n 1/^nC_r ∑_r=0^n n r/^nC_r=na_n ∑_r=0^n n r/^nC_n r [∵ ^nC_r = ...

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Byju's Answer

Standard XI

Mathematics

Binomial Coefficients

If a n =∑ r =...

Question

If $a_{n} = \sum_{r = 0}^{n} \frac{1}{^{n} C_{r}}$ then $\sum_{r = 0}^{n} \frac{r}{^{n} C_{r}}$ equals

$(n - 1) a_{n}$

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$n a_{n}$

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$\frac{1}{2} n a_{n}$

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None of these

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Solution

The correct option is C
$\frac{1}{2} n a_{n}$

Let $b = \sum_{r = 0}^{n} \frac{r}{^{n} C_{r}} = \sum_{r = 0}^{n} \frac{n - (n - r)}{^{n} C_{r}}$
$= n \sum_{r = 0}^{n} \frac{1}{^{n} C_{r}} - \sum_{r = 0}^{n} \frac{n - r}{^{n} C_{r}} = n a_{n} - \sum_{r = 0}^{n} \frac{n - r}{^{n} C_{n - r}} [∵^{n} C_{r} =^{n} C_{n - r}] = n a_{n} - b \Rightarrow 2 b = n a_{n} \Rightarrow b = \frac{n}{2} a_{n}$

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If an=∑nr=01nCr then ∑nr=0rnCr equals

The correct option is C 12nan Let b=∑nr=0rnCr=∑nr=0n−(n−r)nCr =n∑nr=01nCr−∑nr=0n−rnCr=nan−∑nr=0n−rnCn−r[∵ nCr=nCn−r]=nan−b⇒ 2b=nan ⇒ b=n2an

If $a_{n} = \sum_{r = 0}^{n} \frac{1}{^{n} C_{r}}$ then $\sum_{r = 0}^{n} \frac{r}{^{n} C_{r}}$ equals