If the fourth term in the binomial expansion of [( 2 x + x^ _8x )^6] (x 0) is 20 87, the…

Question

If the fourth term in the binomial expansion of $${\left( {{2 \over x} + {x^{{{\log }_8}x}}} \right)^6}$$
 (x > 0) is 20 \times  87, then a value of
x is :

Accepted Answer

$${\left( {{2 \over x} + {x^{{{\log }_8}x}}} \right)^6}$$

Given T4 = 20 \times  87

$$ \Rightarrow $$ $${}^6{C_3}{\left( {{2 \over x}} \right)^3}{\left( {{x^{{{\log }_8}x}}} \right)^3}$$ = 20 \times  87

$$ \Rightarrow $$ 20$$ \times $$ ${8 \over {{x^3}}} \times {x^{3{{\log }_8}x}}$$ = 20 \times  87

$$ \Rightarrow $${x^{3{{\log }_8}x - 3}}$= 86

Taking${{{\log }_8}}$both side$ \Rightarrow $(${3{{\log }_8}x - 3}$)$ \times $${{{\log }_8}x}$= 6$ \Rightarrow $$3{\left( {{{\log }_8}x} \right)^2}$- 3${{{\log }_8}x}$= 6$ \Rightarrow $${\left( {{{\log }_8}x} \right)^2}$-${{{\log }_8}x}$= 2$ \Rightarrow $(${{{\log }_8}x}$- 2)(${{{\log }_8}x}$+ 1) = 0$ \Rightarrow $${{{\log }_8}x}$= 2 or${{{\log }_8}x}$= -1$ \Rightarrow $x = 82 or x =${1 \over 8}$

Solutions · 1