The coefficient of the middle term in the binomial expansion in powers of x of (1 +α x)4 and of (1 - α x)6 is the same, then -α equals to

Question

The coefficient of the middle term in the binomial expansion in powers of x of (1 +α x)4 and of (1 - α x)6 is the same, then -α equals to (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

The middle term in the expansion of (1 + α x)4 = T3 = 4C2(α x)2 = 6α2 x2 The middle term in the expansion of (1 - α x)6 = T4 = 6C3(-α x)3 = -20α3 x3 According to the question 6α2 = - 20 α3 ⇒ α = -(3/10)

Q. The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1 + αx)^{4}$ and of $(1 - αx)^{6}$ is the same, then $- α$ equals to

The middle term in the expansion of
$(1 + αx)^{4} = T_{3} =^{4} C_{2} (αx)^{2} = 6 α^{2} x^{2}$
The middle term in the expansion of
$(1 - αx)^{6} = T_{4} =^{6} C_{3} (- αx)^{3} = - 20 α^{3} x^{3}$
According to the question
$6 α^{2} = - 20 α^{3}$
$\Rightarrow α = - \frac{3}{10}$

Q. The coefficient of the middle term in the binomial expansion in powers of x of (1+αx)4 and of (1−αx)6 is the same, then −α equals to

The middle term in the expansion of (1+αx)4=T3​=4C2​(αx)2=6α2x2 The middle term in the expansion of (1−αx)6=T4​=6C3​(−αx)3=−20α3x3 According to the question 6α2=−20α3 ⇒α=−103​

Q. The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1 + αx)^{4}$ and of $(1 - αx)^{6}$ is the same, then $- α$ equals to

The middle term in the expansion of
$(1 + αx)^{4} = T_{3} =^{4} C_{2} (αx)^{2} = 6 α^{2} x^{2}$
The middle term in the expansion of
$(1 - αx)^{6} = T_{4} =^{6} C_{3} (- αx)^{3} = - 20 α^{3} x^{3}$
According to the question
$6 α^{2} = - 20 α^{3}$
$\Rightarrow α = - \frac{3}{10}$