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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 :
$\frac{{^m{C_t}\,\,\left( {{p^t}\, - \,{q^t}} \right)}}{{p\, - \,q}}$
$\frac{{^m{C_t}\,\,\left( {{p^{m\, - \,t}}\, - \,{q^{m\, - \,t}}} \right)}}{{p\, - \,q}}$
$\frac{{^m{C_t}\,\,\left( {{p^t}\, + \,{q^t}} \right)}}{{p\, - \,q}}$
$\frac{{^m{C_t}\,\,\left( {{p^{m\, - \,t}}\, + \,{q^{m\, - \,t}}} \right)}}{{p\, - \,q}}$
Solution
$E = (\alpha + p)^{m – 1}$ $\left[ {1\,\, + \,\,\frac{{\alpha \, + \,q}}{{\alpha \, + \,p}}\,\, + \,\,{{\left( {\frac{{\alpha \, + \,q}}{{\alpha \, + \,p}}} \right)}^2}\,\, + \,\,……\,\, + \,\,{{\left( {\frac{{\alpha \, + \,q}}{{\alpha \, + \,p}}} \right)}^{m\, – \,1}}} \right]$
$\Rightarrow$ co-efficient of $\alpha ^t$ $= \frac{{{{\left( {\alpha \, + \,p} \right)}^m}\,\, – \,\,{{\left( {\alpha \, + \,q} \right)}^m}}}{{p\,\, – \,\,q}}$ or $\frac{{{{\left( {p\, + \,\alpha } \right)}^m}\,\, – \,\,{{\left( {q\, + \,\alpha } \right)}^m}}}{{p\,\, – \,\,q}}$ $=$ $\frac{{^m{C_t}\,\,\left( {{p^{m\, – \,t}}\,\, – \,\,{q^{m\, – \,t}}} \right)}}{{p\, – \,q}}$
