If - r -0 n r -2- r -1 n C r -28-1-6 then n isA- 8B- 4C- 6D- 5

The correct option is D 5 ∑_r=0^n ( r+1/r+1+1/r+1 ) ^nC_r=∑_r=0^n ^nC_r+∑_r=0^n^nC_r/r+1 =2^n+1/n+1∑_r=0^n ^n+1C_r+1 =2^n+2^n+1 1/n+1=(n+3)2^n 1/n+1 =2^8 1/ ...

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Byju's Answer

Standard XII

Mathematics

Greatest Binomial Coefficients

If ∑ r =0 n r...

Question

If $\sum_{r = 0}^{n} (\frac{r + 2}{r + 1})^{n} C_{r} = \frac{2^{8} - 1}{6}$ then n is

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Solution

The correct option is D
5

$\sum_{r = 0}^{n} (\frac{r + 1}{r + 1} + \frac{1}{r + 1})^{n} C_{r} = \sum_{r = 0}^{n}^{n} C_{r} + \sum_{r = 0}^{n} \frac{^{n} C_{r}}{r + 1} = 2^{n} + \frac{1}{n + 1} \sum_{r = 0}^{n}^{n + 1} C_{r + 1} = 2^{n} + \frac{2^{n + 1} - 1}{n + 1} = \frac{(n + 3) 2^{n} - 1}{n + 1} = \frac{2^{8} - 1}{6} \Rightarrow n = 5$

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Similar questions

Q. If

n \sum r = 0 (\frac{r + 2}{r + 1}) C_{r} = \frac{2^{8} - 1}{6}

, where

C_{r} =^{n} C_{r}

, then the value of

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Q. If

n \sum r = 0 (\frac{r + 2}{r + 1})^{n} C_{r} = \frac{2^{8} - 1}{6}

, then

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Q. If

n \sum r = 0 {(\frac{r + 2}{r + 1})}^{n} C_{r} = \frac{2^{8} - 1}{6}

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Q. Statement-1:

\sum_{r = 0}^{n} (r + 1)^{n} C_{r} = (n + 2) 2^{n - 1} .

Statement-2:

\sum_{r = 0}^{n} (r + 1)^{n} C_{r} x^{r} = (1 + x)^{n} + n x (1 + x)^{n - 1} .

Q. Statement-l:

n \sum r = 0 (r + 1)^{n} C_{r} = (n + 2) 2^{n - 1}

Statement-2:

n \sum r = 0 (r + 1)^{n} C_{r} x^{r} = (1 + x)^{n} + n x (1 + x)^{n - 1}

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If ∑nr=0(r+2r+1) nCr=28−16 then n is

The correct option is D 5 ∑nr=0(r+1r+1+1r+1) nCr=∑nr=0 nCr+∑nr=0nCrr+1=2n+1n+1∑nr=0 n+1Cr+1=2n+2n+1−1n+1=(n+3)2n−1n+1=28−16⇒n=5

If $\sum_{r = 0}^{n} (\frac{r + 2}{r + 1})^{n} C_{r} = \frac{2^{8} - 1}{6}$ then n is