The coefficient of the middle term in the bin | vedclass

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Question

(C) The middle term in the expansion of $(1 + \alpha x)^n$ is the $(\frac{n}{2} + 1)^{	ext{th}}$ term.For $(1 + \alpha x)^4$,the middle term is the $

Vedclass · Accepted Answer

$-\frac{3}{10}$

Vedclass · Answer

(C) The middle term in the expansion of $(1 + \alpha x)^n$ is the $(\frac{n}{2} + 1)^{	ext{th}}$ term.For $(1 + \alpha x)^4$,the middle term is the $(\frac{4}{2} + 1)^{	ext{th}} = 3^{	ext{rd}}$ term.The coefficient of the $3^{	ext{rd}}$ term is given by $^4C_2 \alpha^2 = 6 \alpha^2$.For $(1 - \alpha x)^6$,the middle term is the $(\frac{6}{2} + 1)^{	ext{th}} = 4^{	ext{th}}$ term.The coefficient of the $4^{	ext{th}}$ term is given by $^6C_3 (-\alpha)^3 = -20 \alpha^3$.Given that the coefficients are equal:$6 \alpha^2 = -20 \alpha^3$Since $\alpha 
eq 0$,we can divide by $2 \alpha^2$:$3 = -10 \alpha$$\alpha = -\frac{3}{10}$

The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1 + \alpha x)^4$ and of $(1 - \alpha x)^6$ is the same if $\alpha$ equals

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