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Question Number 55786 by gunawan last updated on 04/Mar/19

If the third term in the expansion of   ((1/x) + x^(log_(10) x) )^5  is 1000, then the value of  x is

$$\mathrm{If}\:\mathrm{the}\:\mathrm{third}\:\mathrm{term}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\: \\ $$$$\left(\frac{\mathrm{1}}{{x}}\:+\:{x}^{\mathrm{log}_{\mathrm{10}} {x}} \right)^{\mathrm{5}} \:\mathrm{is}\:\mathrm{1000},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$${x}\:\mathrm{is} \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 04/Mar/19

5c_2 (x)^3 (x^(log_(10) x) )^2 =10^3   x^(3+2log_(10) x) =((1000)/(5!))×2!×3!  x^(3+log_(10) x^2 ) =((1000×2×3×2)/(5×4×3×2))=10^2   (10^n )^(3+log_(10) 10^(2n) ) =10^2   (10^n )^(3+2n) =10^2   3n+2n^2 =2  2n^2 +3n−2=0  2n^2 +4n−n−2=0  2n(n+2)−1(n+2)=0  (n+2)(2n−1)=0  n=−2 and n=(1/2)  so x=10^(−2) =(1/(100))  or x=10^(1/2) =(√(10))

$$\mathrm{5}{c}_{\mathrm{2}} \left({x}\right)^{\mathrm{3}} \left({x}^{{log}_{\mathrm{10}} {x}} \right)^{\mathrm{2}} =\mathrm{10}^{\mathrm{3}} \\ $$$${x}^{\mathrm{3}+\mathrm{2}{log}_{\mathrm{10}} {x}} =\frac{\mathrm{1000}}{\mathrm{5}!}×\mathrm{2}!×\mathrm{3}! \\ $$$${x}^{\mathrm{3}+{log}_{\mathrm{10}} {x}^{\mathrm{2}} } =\frac{\mathrm{1000}×\mathrm{2}×\mathrm{3}×\mathrm{2}}{\mathrm{5}×\mathrm{4}×\mathrm{3}×\mathrm{2}}=\mathrm{10}^{\mathrm{2}} \\ $$$$\left(\mathrm{10}^{{n}} \right)^{\mathrm{3}+{log}_{\mathrm{10}} \mathrm{10}^{\mathrm{2}{n}} } =\mathrm{10}^{\mathrm{2}} \\ $$$$\left(\mathrm{10}^{{n}} \right)^{\mathrm{3}+\mathrm{2}{n}} =\mathrm{10}^{\mathrm{2}} \\ $$$$\mathrm{3}{n}+\mathrm{2}{n}^{\mathrm{2}} =\mathrm{2} \\ $$$$\mathrm{2}{n}^{\mathrm{2}} +\mathrm{3}{n}−\mathrm{2}=\mathrm{0} \\ $$$$\mathrm{2}{n}^{\mathrm{2}} +\mathrm{4}{n}−{n}−\mathrm{2}=\mathrm{0} \\ $$$$\mathrm{2}{n}\left({n}+\mathrm{2}\right)−\mathrm{1}\left({n}+\mathrm{2}\right)=\mathrm{0} \\ $$$$\left({n}+\mathrm{2}\right)\left(\mathrm{2}{n}−\mathrm{1}\right)=\mathrm{0} \\ $$$${n}=−\mathrm{2}\:{and}\:{n}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${so}\:{x}=\mathrm{10}^{−\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{100}} \\ $$$${or}\:{x}=\mathrm{10}^{\frac{\mathrm{1}}{\mathrm{2}}} =\sqrt{\mathrm{10}}\: \\ $$$$ \\ $$$$ \\ $$

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