Question

Find the first three terms in the expansion of $(1+3x{)}^{\frac{1}{2}}(1-2x{)}^{-\frac{1}{3}}$

Answer 1

Whatever be value of $n$, negative integer or positive or negative fraction,

$(1+x{)}^n=1+\frac{n}{1!}x+\frac{n(n-1)}{2!}{x}^2+\frac{n(n-1)(n-2)}{3!}{x}^3+...+...+\frac{n(n-1)(n-2)...(n-r+1)}{r!}{x}^r+...$

where, $|x|<1$ i.e. $-1<x<1.$

Now,

Expanding the two binomials as far as the term containing $x^2$ using above formula, we have

$=(1+\frac{3}{2}x-\frac{9}{8}{x}^2-.........)(1+\frac{2}{3}x+\frac{8}{9}{x}^2+............)$

$=1+x(\frac{3}{2}+\frac{2}{3})+x^2[-\frac{9}{8}+(\frac{3}{2}\cdot \frac{2}{3})+\frac{8}{9}]+....$

$=1+\frac{13}{6}x+\frac{55}{72}{x}^2$