Find the coefficient of alpha^t in the expans | vedclass

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(B) The given expression is a geometric series with $m$ terms,where the first term $a = (\alpha + p)^{m - 1}$ and the common ratio $r = \frac{\alpha +

Vedclass · Accepted Answer

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

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(B) The given expression is a geometric series with $m$ terms,where the first term $a = (\alpha + p)^{m - 1}$ and the common ratio $r = \frac{\alpha + q}{\alpha + p}$.Using the sum formula $S_m = a \frac{1 - r^m}{1 - r}$,we get:$E = (\alpha + p)^{m - 1} \left[ \frac{1 - (\frac{\alpha + q}{\alpha + p})^m}{1 - \frac{\alpha + q}{\alpha + p}} \right] = (\alpha + p)^{m - 1} \left[ \frac{\frac{(\alpha + p)^m - (\alpha + q)^m}{(\alpha + p)^m}}{\frac{\alpha + p - \alpha - q}{\alpha + p}} \right] = \frac{(\alpha + p)^m - (\alpha + q)^m}{p - q}$.Now,we find the coefficient of $\alpha^t$ in the expansion of $\frac{(\alpha + p)^m - (\alpha + q)^m}{p - q}$.The coefficient of $\alpha^t$ in $(\alpha + p)^m$ is $^mC_t p^{m - t}$ and in $(\alpha + q)^m$ is $^mC_t q^{m - t}$.Thus,the coefficient of $\alpha^t$ in the expression is $\frac{^mC_t p^{m - t} - ^mC_t q^{m - t}}{p - q} = \frac{^mC_t (p^{m - t} - q^{m - t})}{p - q}$.

Find the coefficient of $\alpha^t$ in the expansion of $(\alpha + p)^{m - 1} + (\alpha + p)^{m - 2} (\alpha + q) + (\alpha + p)^{m - 3} (\alpha + q)^2 + \dots + (\alpha + q)^{m - 1}$,where $\alpha \neq -q$ and $p \neq q$.

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