If -r-02narx-2r-r-02nbrx-3r and ak-1 for all k- n - then bn is equal to

Clearly, b_n is the coefficient of (x 3)^n in the expression ∑_r=0^2nb_r (x 3)^r. ∴ b_n=coefficient of (x 3)^n in (∑_r=0^2na_r(x 2)^r) ⋯ (1) nbsp; nbsp; ...

If ∑r=02narx-...

Question

If $2 n \sum r = 0 a_{r} (x - 2)^{r} = 2 n \sum r = 0 b_{r} (x - 3)^{r}$ and $a_{k} = 1$ for all $k \geq n,$ then $b_{n}$ is equal to

^{2 (n + 1)} C_{2 n + 1}

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^{2 n + 1} C_{n + 1}

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^{2 n + 1} C_{n + 2}

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None of these

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Solution

The correct option is B $^{2 n + 1} C_{n + 1}$
Clearly, $b_{n}$ is the coefficient of $(x - 3)^{n}$ in the expression $2 n \sum r = 0 b_{r} (x - 3)^{r} .$

$∴ b_{n} = coefficient of (x - 3)^{n} in (2 n \sum r = 0 a_{r} (x - 2)^{r}) \dots (1)$

$= coefficient of (x - 3)^{n} in n - 1 \sum r = 0 a_{r} (x - 2)^{r} + 2 n \sum r = n a_{r} (x - 2)^{r}$

$(∵ a_{k} = 1$ for all $k \geq n)$

$= coefficient of (x - 3)^{n} in 2 n \sum r = n (x - 2)^{r}$

$= coefficient of (x - 3)^{n} in [(x - 2)^{n} {\frac{(x - 2)^{n + 1} - 1}{(x - 2) - 1}}]$

$= coefficient of (x - 3)^{n} in (\frac{(x - 2)^{2 n + 1} - (x - 2)^{n}}{x - 3})$

$= coefficient of (x - 3)^{n + 1} in {(x - 2)^{2 n + 1} - (x - 2)^{n}}$

$= coefficient of (x - 3)^{n + 1} in (x - 2)^{2 n + 1}$

$= coefficient of (x - 3)^{n + 1} in [(x - 3) + 1]^{2 n + 1}$

$= coefficient of (x - 3)^{n + 1} in {2 n + 1 \sum r = 0^{2 n + 1} C_{r} (x - 3)^{r}}$

$=^{2 n + 1} C_{n + 1}$

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If $2 n \sum r = 0 a_{r} (x - 2)^{r} = 2 n \sum r = 0 b_{r} (x - 3)^{r}$ and $a_{k} = 1$ for all $k \geq n,$ then $b_{n}$ is equal to