∑ r = 0 n n − r + 1 n − 3 r + 1 2 r n C r is equal to

Let S = ∑ r = 0 n n − r + 1 n − 3 r + 1 2 r n C r . Rewrite as S = ∑ r = 0 n n − r + 1 n + 1 2 r n C r − 3 ∑ r = 0 n n − r + 1 r 2 r n C r . After simplification, S = 2 n 1 . Therefore, option A is correct.

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Mathematics - Binomial Theorem Question with Solution | TestHub

MathematicsBinomial TheoremProperties of Binomial coefficientsMedium2 minQB

MathematicsMediumsingle choice

$\sum_{r = 0}^{n} \frac{n - 3 r + 1}{n - r + 1} \frac{^{n} C _{r}}{2 ^{r}}$ is equal to

Options:

Answer:

Solution:

Let $S = \sum_{r = 0}^{n} \frac{n - 3 r + 1}{n - r + 1} \frac{^{n} C _{r}}{2 ^{r}}$ . Rewrite as $S = \sum_{r = 0}^{n} \frac{n + 1}{n - r + 1} \frac{^{n} C _{r}}{2 ^{r}} - 3 \sum_{r = 0}^{n} \frac{r}{n - r + 1} \frac{^{n} C _{r}}{2 ^{r}}$ . After simplification, $S = \frac{1}{2 ^{n}}$ .

Therefore, option A is correct.

Stream:JEESubject:MathematicsTopic:Binomial TheoremSubtopic:Properties of Binomial coefficients

⏱ 2mℹ️ Source: QB

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