The coefficient of the middle term in the binomial expansion in powers of x of (1+ x)^4 an…

Question

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:

Accepted Answer

For $${\left( {1 + \alpha x} \right)^4}$$ the middle term $${T_{{4 \over 2} + 1}}$$ = $${}^4{C_2}.{\alpha ^2}{x^2}$$

$$\therefore$$ Coefficient of middle term = $${}^4{C_2}.{\alpha ^2}$$

For $${\left( {1 - \alpha x} \right)^6}$$ the middle term $${T_{{6 \over 2} + 1}}$$ = $${}^6{C_3}.-{\alpha ^3}{x^3}$$

$$\therefore$$ Coefficient of middle term = $${}^6{C_3}.{-\alpha ^3}$$

$$\therefore$$ According to question,

$${}^4{C_2}.{\alpha ^2}$$ = $${}^6{C_3}.{-\alpha ^3}$$

$$ \Rightarrow 6 = 20 \times  - \alpha $$

$$ \Rightarrow \alpha  =  - {3 \over {10}}$$

Solutions · 1