Question

Using Binomial Theorem expand $\frac{4+x}{\sqrt{(1+4x)}}$ till the term containing $x^3$.

Answer 1

Expand $(4+x){(1+4x)}^{\frac{-1}{2}}$ upto $x^3$

$=(4+x)[1+(\frac{-1}{2}).(4x)+\frac{\frac{-1}{2}.\frac{-3}{2}}{2!}.{\left(4x\right)}^2+.....+\frac{\frac{-1}{2}.\frac{-3}{2}.\frac{-5}{2}}{3!}.{\left(4x\right)}^3]$

$=(4+x)[1-2x+{6x}^2-{20x}^3]$

$=4-8x+{24x}^2-{80x}^3+{x-2x}^2+{6x}^3-{20x}^4$

$=4-7x+{22x}^2-{74x}^3..........$