The value of $$\frac{30_{C_1}}{2} + \frac{30_{C_3}}{4} + \frac{30_{C_5}}{6} +...+ \frac{30_{C_{29}}}{30}$$ is

We know that $$(1+x)^n$$ = $$^nC_0*1^n+^nC_1*1^{n-1}*x......................................^nC_n*x^n$$

On Integrating, we get

$$\frac{(1+x)^{n+1}}{n+1}$$ = $$\frac{^nC_0*1^n}{1}+\frac{^nC_1*1^{n-1}*x}{2}......................................\frac{^nC_n*x^n}{n+1}$$

When x = 1

$$\frac{2^{31}}{31}$$ =  $$\frac{30_{C_0}}{1}$$ +$$\frac{30_{C_1}}{2} + \frac{30_{C_2}}{3} +\frac{30_{C_3}}{4} + \frac{30_{C_5}}{6} +...+ \frac{30_{C_{29}}}{30} + \frac{30_{C_{30}}}{31}$$   -- Eq 1

When x = -1

0= $$ \frac{30_{C_0}}{1}$$ -$$\frac{30_{C_1}}{2} + \frac{30_{C_2}}{3} -\frac{30_{C_3}}{4} + \frac{30_{C_5}}{6} +...+ \frac{30_{C_{29}}}{30} - \frac{30_{C_{30}}}{31}$$  -- Eq 2

Adding Eq 1 and 2, we get

$$\frac{2^{31}}{31}$$ = 2($$\frac{30_{C_1}}{2} + \frac{30_{C_3}}{4} + \frac{30_{C_5}}{6} +...+ \frac{30_{C_{29}}}{30}$$)

($$\frac{30_{C_1}}{2} + \frac{30_{C_3}}{4} + \frac{30_{C_5}}{6} +...+ \frac{30_{C_{29}}}{30}$$) = $$\frac{2^{30}}{31}$$