Question

Co-efficient of ${\alpha }^t$ in the expansion of,
$(\alpha +p{)}^{m-1}+(\alpha +p{)}^{m-2}(\alpha +q)+(\alpha +p{)}^{m-3}(\alpha +q{)}^2+........(\alpha +q{)}^{m-1},$ where $\alpha \ne -q$ and $p\ne q$ is $(\alpha +p{)}^{m-1}+(\alpha +p{)}^{m-2}(\alpha +q)+(\alpha +p{)}^{m-3}(\alpha +q{)}^2+........(\alpha +q{)}^{m-1},$

• A. $\frac{{}^m{C}_t(p^t-q^t)}{p-q}$
• B. $\frac{{}^m{C}_t(p^{m-t}-q^{m-t})}{p-q}$
• C. $\frac{{}^m{C}_t(p^t+q^t)}{p-q}$
• D. $\frac{{}^m{C}_t(p^{m-t}+q^{m-t})}{p-q}$.

Answer 1

Answer: (B)

$\frac{{}^m{C}_t(p^{m-t}-q^{m-t})}{p-q}$

Let $S=(\alpha +p{)}^{m-1}+(\alpha +p{)}^{m-2}(\alpha +q)+(\alpha +p{)}^{m-3}(\alpha +q{)}^2+....+(\alpha +q{)}^{m-1}$

As we can see, $S$ is a Geometric Sum with,

First term, $a=(\alpha +p{)}^{m-1}$,

Common Ratio, $r=\frac{\alpha +q}{\alpha +p}$ and

Number of terms, $n=m$

So, $S_m=\frac{(\alpha +p{)}^{m-1}(1-{\left(\frac{\alpha +q}{\alpha +p}\right)}^m)}{(1-\frac{\alpha +q}{\alpha +p})}$

$\Rightarrow S_m=\frac{(\alpha +q{)}^m-(\alpha +q{)}^m}{p-q}$

On doing binomial expansion on $S_m$,

$S_m=\frac{1}{p-q}({\sum }_{t=0}^m{}^m{C}_t{\alpha }^t{p}^{m-t}-{\sum }_{t=0}^m{}^m{C}_t{\alpha }^t{q}^{m-t})$

From the equation, we can see that,

Coefficient of ${\alpha }^t=\frac{{}^m{C}_t(p^{m-t}-q^{m-t})}{p-q}$

So, Option B is the correct answer.