The value of - r -0 n -1 n C r - n C r - n C r -1 equalsA- n-2B- n-1C- None of theseD- n-2

The correct option is A n/2∑_r=0^n 1^n C_r/^n C_r + ^n C_r+1 = ∑_r=0^n 11/1+^n C_r+1/^n C_r = ∑_r=0^n 11/1+n r/r+1 =∑_r=0^n 1r+1/n+1=1/n+1∑_r=0^n 1(r+1) = 1/(n ...

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Standard XII

Mathematics

Greatest Binomial Coefficients

The value of ...

Question

The value of $\sum_{r = 0}^{n - 1} \frac{^{n} C_{r}}{^{n} C_{r} +^{n} C_{r + 1}}$ equals

n + 1

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\frac{n}{2}

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n + 2

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None of these

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Solution

The correct option is B $\frac{n}{2}$
$\sum_{r = 0}^{n - 1} \frac{^{n} C_{r}}{^{n} C_{r} +^{n} C_{r + 1}} = \sum_{r = 0}^{n - 1} \frac{1}{1 + \frac{^{n} C_{r + 1}}{^{n} C_{r}}} = \sum_{r = 0}^{n - 1} \frac{1}{1 + \frac{n - r}{r + 1}}$
$= \sum_{r = 0}^{n - 1} \frac{r + 1}{n + 1} = \frac{1}{n + 1} \sum_{r = 0}^{n - 1} (r + 1) = \frac{1}{(n + 1)} [1 + 2 + \dots + n] = \frac{n}{2}$

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Similar questions

Find the value of $\sum_{r = 0}^{n - 1} \frac{^{n} C_{r}}{^{n} C_{r} +^{n} C_{r + 1}}$ when n=100

Q. The value of

n - 1 \sum r = 0 \frac{^{n} C_{r}}{^{n} C_{r} +^{n} C_{r + 1}}

is equals to ..............

Q. The value of

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

equals

Q. Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
(a)

\frac{{}^{n}C_{r}}{{}^{n}C_{r - 1}} = \frac{n - r + 1}{r}

(b) n ·_{^{n −}}_¹C_r_{− 1} = (n − r + 1) ⁿC_r_{− 1}
(c)

\frac{{}^{n}C_{r}}{{}^{n - 1}C_{r - 1}} = \frac{n}{r}

(iv) ⁿC_r + 2 · ⁿC_r_{− 1} + ⁿC_r_{− 2} = ^{n +}²C_r.

\sum_{r = 0}^{n - 1} \frac{^{n} c_{r}}{^{n} c_{r} +^{n} c_{r} + i}

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The value of ∑n−1r=0nCrnCr+nCr+1 equals

The correct option is B n2∑n−1r=0nCrnCr+nCr+1=∑n−1r=011+nCr+1nCr=∑n−1r=011+n−rr+1 =∑n−1r=0r+1n+1=1n+1∑n−1r=0(r+1)=1(n+1)[1+2+…+n]=n2

The value of $\sum_{r = 0}^{n - 1} \frac{^{n} C_{r}}{^{n} C_{r} +^{n} C_{r + 1}}$ equals