If Ar is the coefficient of xr in the expansion of 1-x21-x-221-x-2221-x-232- prove that Ar-22-2r-1Ar-1-Ar-2- and A4-1072-315-

⇒( 1+x ) #160; ^ 2 ( 1+ x 2 #160; ) #160; ^ 2 ( 1+ x 2 ^ 2 #160; ) #160; ^ 2 .... =1+ A _ 1 x+ A _ 2 x ^ 2 +... Change x into x 2 ...

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Standard XII

Mathematics

Greatest Binomial Coefficients

If Ar is th...

Question

If $A_{r}$ is the coefficient of $x^{r}$ in the expansion of $(1 + x)^{2} {(1 + \frac{x}{2})}^{2} {(1 + \frac{x}{2^{2}})}^{2} {(1 + \frac{x}{2^{3}})}^{2} . . . . . .,$ prove that $A_{r} = \frac{2^{2}}{2^{r} - 1} (A_{r - 1} + A_{r - 2}),$ and $A_{4} = \frac{1072}{315}$ .

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Solution

$\Rightarrow {(1 + x)}^{2} {(1 + \frac{x}{2})}^{2} {(1 + \frac{x}{2^{2}})}^{2} . . . .$
$= 1 + A_{1} x + A_{2} x^{2} + . . .$
Change x into $\frac{x}{2}$
${(1 + \frac{x}{2})}^{2} {(1 + \frac{x}{2^{2}})}^{2} {(1 + \frac{x}{2^{3}})}^{2} . . . .$
$= 1 + A_{1} \frac{x}{2} + A_{2} \frac{x^{2}}{2^{2}} + . . . + A_{r} \frac{x^{r}}{r^{r}} + . . . . .$
${(1 + x)}^{2} (1 + A_{1} \frac{x}{2} + A_{2} \frac{x^{2}}{2^{2}} + . . .) = 1 + A_{1} x + A_{2} x^{2} + . . .$
$A_{0} = 0; A_{1} =$ Sum of co efficient of 'x' in the component factor
$= 2 (1 + \frac{1}{2} + \frac{1}{2^{2}} + . . . . .) = 4$
Also;-
$A_{r} = \frac{4 (A_{r - 1} + A_{r - 2})}{2^{r} - 1}$
Putting $r = 2$
$A_{2} = \frac{4}{3} (4 + 3) = \frac{20}{3}$
$A_{3} = \frac{4}{7} (20 + 4) = \frac{128}{21}$
$A_{4} = \frac{4}{15} (\frac{128}{21} + \frac{20}{3}) = \frac{1072}{315}$

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If Ar is the coefficient of xr in the expansion of (1+x)2(1+x2)2(1+x22)2(1+x23)2......, prove that Ar=222r−1(Ar−1+Ar−2), and A4=1072315.

If $A_{r}$ is the coefficient of $x^{r}$ in the expansion of $(1 + x)^{2} {(1 + \frac{x}{2})}^{2} {(1 + \frac{x}{2^{2}})}^{2} {(1 + \frac{x}{2^{3}})}^{2} . . . . . .,$ prove that $A_{r} = \frac{2^{2}}{2^{r} - 1} (A_{r - 1} + A_{r - 2}),$ and $A_{4} = \frac{1072}{315}$ .