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Binomial Theorem for Any Index

Find the coef...

Question

Find the coefficient of x in the expansion of $(1 - 2 x^{3} + 3 x^{5}) {(1 + \frac{1}{x})}^{8}$

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Solution

efficient of $x$ in $(1 - 2 x^{3} + 3 x^{5}) {(1 + \frac{1}{x})}^{8}$
$∴$ coefficient of $x$ in $(1 + \frac{1}{x})$ +coefficient of $x$ in $- 2 x^{2} {(1 + \frac{1}{x})}^{8}$ + coefficient of $x$ in $(3 x^{5}) {(1 + \frac{1}{x})}^{8}$
= coefficient of $x$ in ${(1 + \frac{1}{x})}^{8} - 2$ coefficient of $x^{- 2}$ in ${(1 + \frac{1}{x})}^{7} + 3$
$3$ coefficient of $n^{- 4}$ in ${(1 + \frac{1}{x})}^{8}$
$r^{t h}$ term in ${(1 + \frac{1}{x})}^{8}$
$t_{r} =^{8} C_{r - 1} {(\frac{1}{x})}^{r - 1} =^{8} C_{r - 1} x^{1 - r}$
for $1 - r = 1 r = 0,$ coefficient of $x$ in $(1 + \frac{1}{x}) = 0$
for $1 - r = - 2$ i.e. $r = 3$ coefficient of $x$ in ${(1 + \frac{1}{x})}^{8} =^{8} C_{2}$
for $1 - r = - 4$ i,e, $r = 5$ coefficient of $x$ in ${(1 + \frac{1}{x})}^{8} =^{8} C_{5}$
$∴ 0 - 2 (^{8} C_{2}) + 3 (^{8} C_{4}) = 154$

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