- r -02 n a r x -2 r - r -02 n b r x -3 r and a k -1 for all k - n- then b n is equal toB- 2 n -1 C n -2C- 2n-1 C2 n-1D- None of these

The correct option is A ^2n+1C_n+1Clearly, 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=0^2n ...

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Byju's Answer

Standard XII

Mathematics

Binomial Expression

∑ r =02 n a r...

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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Similar questions

Q. If

\sum_{r = 0}^{2 n} a_{r} (x - 2)^{2} = \sum_{i = 0}^{2 n} b_{r} (x - 3)^{r} a n d a_{k} = 1

for all

k \geq n

, then show

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

Q. Find the value of

^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + . . . . . . +^{2 n + 1} C_{n} +^{2 n + 1} C_{n + 1} +^{2 n + 1}

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

n = 3

___

The value of $C_{0} + 3 C_{1} + 5 C_{2} + 7 C_{3} + . . . . + (2 n + 1) C_{n}$ is equal to :

Q. IF

S_{n} = \sum_{r = 0}^{n} \frac{1}{^{n} C_{r}}

and

t_{n} = \sum_{r = 0}^{n} \frac{r}{^{n} C_{r}},

then

\frac{t_{n}}{S_{n}}

is equal to

Q. If

\sum_{r = 0}^{2 n} a_{r} {(x - 100)}^{r} = \sum_{r = 0}^{2 n} b_{r} {(x - 101)}^{r}

and

a_{k} = \frac{2^{k}}{{^{k} C}_{n}}

for all

k \geq n

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b_{n}

equals

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If 2n∑r=0ar(x−2)r=2n∑r=0br(x−3)r and ak=1 for all k≥n, then bn is equal to

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