Question

If $|x|$<2/3, then what is the 4th term in the expansion ${(1+\frac{3}{2}x)}^{1/2}$?

Answer 1

$\to (1+\frac{3}{2}x{)}^{\frac{1}{2}}=1+(\frac{1}{2}\left)\right(\frac{3x}{2})+\frac{\left(\frac{1}{2}\right)\left(\frac{-1}{2}\right)}{2}(\frac{3x}{2}{)}^2$

$+\frac{\left(\frac{1}{2}\right)\left(\frac{-1}{2}\right)\left(\frac{-3}{2}\right)}{6}(\frac{3x}{2}{)}^3+...$

$\Rightarrow$ Forth term $=\frac{3}{8\times 6}\times \frac{27x^3}{8}$

$=\frac{81}{512}{x}^3$