Question

The value of 30C1/2 + 30C3/4 + 30C5/6 + ….. + 30B29/30 is

Answer 1

$\frac{^{30}\textrm{C}_{1} }{2} +\frac{^{30}\textrm{C}_{3} }{4} +\frac{^{30}\textrm{C}_{5} }{6} +.........+\frac{^{30}\textrm{C}_{27} }{28}+\frac{^{30}\textrm{C}_{29} }{30}$$=\frac{2^{30}}{31}$

Step-by-step explanation:

we know  that

$(1+x)^{30}$=$^{30}\textrm{c}_{0}+^{30}\textrm{c}_{1}x+^{30}\textrm{c}_{2}x^{2} +...........+^{30}\textrm{c}_{29}x^{29}+^{30}\textrm{c}_{30}x^{30}$

$(1-x)^{30}$=$^{30}\textrm{c}_{0}-^{30}\textrm{c}_{1}x+^{30}\textrm{c}_{2}x^{2} -...........-^{30}\textrm{c}_{29}x^{29}+^{30}\textrm{c}_{30}x^{30}$

Therefore,

$(1+x)^{30}$+$(1-x)^{30}$=$^{30}\textrm{c}_{0}+^{30}\textrm{c}_{1}x+^{30}\textrm{c}_{2}x^{2} +...........+^{30}\textrm{c}_{29}x^{29}+^{30}\textrm{c}_{30}x^{30}$+$^{30}\textrm{c}_{0}-^{30}\textrm{c}_{1}x+^{30}\textrm{c}_{2}x^{2} -...........-^{30}\textrm{c}_{29}x^{29}+^{30}\textrm{c}_{30}x^{30}$

⇒$(1+x)^{30}$-$(1-x)^{30}$=$2(^{30}\textrm{c}_{0}+^{30}\textrm{c}_{2}x^2+^{30}\textrm{c}_{4}x^{4} +...........+^{30}\textrm{c}_{28}x^{28}+^{30}\textrm{c}_{30}x^{30})$

Now integrating both sides with limit 0 to 1

$\int\limits^1_0[{(1+x)^{30}-(1-x)^{30}} ]\, dx=$$\int\limits^1_0 {2(^{30}\textrm{c}_{1}x+^{30}\textrm{c}_{3}x^3+^{30}\textrm{c}_{5}x^{5} +...........+^{30}\textrm{c}_{27}x^{27}+^{30}\textrm{c}_{29}x^{29})} \, dx$

⇒$\left [ \frac{(1+x)^{31}}{31} \right ]^{1}_{0}}$$-\left [ \frac{(1-x)^{31}}{31} \right ]^{1}_{0}}$$=2\left [ \frac{^{30}\textrm{C}_{1} }{2}x^2\right ]^{1}_{0}$$+2\left [ \frac{^{30}\textrm{C}_{3} }{4}x^4\right ]^{1}_{0}$$+2\left [ \frac{^{30}\textrm{C}_{5} }{6}x^6\right ]^{1}_{0}+........+$$+2\left [ \frac{^{30}\textrm{C}_{27} }{28}x^{28}\right ]^{1}_{0}$$+2\left [ \frac{^{30}\textrm{C}_{29} }{30}x^{30}\right ]^{1}_{0}$

⇒$\frac{2^{31}}{31} -\frac{1}{31} +\frac{1}{31}=$$2(\frac{^{30}\textrm{C}_{1} }{2} +\frac{^{30}\textrm{C}_{3} }{4} +\frac{^{30}\textrm{C}_{5} }{6} +.........+\frac{^{30}\textrm{C}_{27} }{28}+\frac{^{30}\textrm{C}_{29} }{30})$

⇒$\frac{^{30}\textrm{C}_{1} }{2} +\frac{^{30}\textrm{C}_{3} }{4} +\frac{^{30}\textrm{C}_{5} }{6} +.........+\frac{^{30}\textrm{C}_{27} }{28}+\frac{^{30}\textrm{C}_{29} }{30}$$=\frac{2^{30}}{31}$