Answer to Question #228764 in Combinatorics | Number Theory for Kamfx

Question

Determine the first 4 terms in the bionomial expansion (1+2x)^-3/4

Answer 1

$$(1+y)^k=1+ky+\dfrac{k(k-1)}{2!}y^2+\dfrac{k(k-1)(k-2)}{3!}y^3+...$$

$$k=-\dfrac{3}{4}, y=2x$$

$$(1+2x)^{-3/4}=1+(-\dfrac{3}{4})(2x)+\dfrac{-\dfrac{3}{4}(-\dfrac{3}{4}-1)}{2!}(2x)^2$$

$$+\dfrac{-\dfrac{3}{4}(-\dfrac{3}{4}-1)(-\dfrac{3}{4}-2)}{3!}(2x)^3+...$$

$$a_4=\dfrac{-\dfrac{3}{4}(-\dfrac{7}{4})(-\dfrac{11}{4})}{3!}(8)x^3$$

$$a_4=-\dfrac{77}{16}x^3$$