If xr occurs in the expansion of x- 1 x n- find its coefficient-

We can write the expression as x^n ( 1+1x^2 #10; )^n which can be written as x^n∑_p=0 #10;^n ^nC_p #160; ( 1x^2)^n p Coefficient of x^n+2p ...

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Standard XII

Mathematics

Binomial Coefficients

If xr occur...

Question

If $x^{r}$ occurs in the expansion of ${(x + \frac{1}{x})}^{n}$ , find its coefficient.

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Solution

We can write the expression as $x^{n} {(1 + \frac{1}{x^{2}})}^{n}$ which can be written as $x^{n} n \sum p = 0^{n} C_{p} {(\frac{1}{x^{2}})}^{n - p}$
Coefficient of $x^{n + 2 p - 2 n}$ is $^{n} C_{p}$
To find coefficient of $x^{r}$ , put $r = n + 2 p - 2 n \Rightarrow p = \frac{r + n}{2}$
$∴$ The coefficient of $x^{r}$ is $^{n} C_{\frac{r + n}{2}}$

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If xr occurs in the expansion of (x+1x)n, find its coefficient.

We can write the expression as xn(1+1x2)n which can be written as xnn∑p=0 nCp(1x2)n−pCoefficient of xn+2p−2n is nCpTo find coefficient of xr, put r=n+2p−2n⇒p=r+n2∴ The coefficient of xr is nCr+n2

If $x^{r}$ occurs in the expansion of ${(x + \frac{1}{x})}^{n}$ , find its coefficient.