Co-efficient of alpha ^t in the expansion of, | vedclass

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Question

$E = (\alpha + p)^{m - 1}$ $\left[ {1\,\, + \,\,\frac{{\alpha \, + \,q}}{{\alpha \, + \,p}}\,\, + \,\,{{\left( {\frac{{\alpha \, + \,q}}{{\alpha \, +

Vedclass · Accepted Answer

$\frac{{^m{C_t}\,\,\left( {{p^{m\, - \,t}}\, - \,{q^{m\, - \,t}}} \right)}}{{p\, - \,q}}$

Vedclass · Answer

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

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 :

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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 :