If `f(x)=(a^(x))/(a^(x)+sqrt(a))(agt0),g(n)=sum_(r=1)^(2n-1)2f((r)/(2n))`. Find te value `g(4)`

Answer

7

Explanation

`f(x)=(a^(x))/(a^(x)+sqrt(a))(agt0)`,,,,.
`f(x)+f(1-x)=(a^(x))/(a^(x)+sqrt(a))+(a^(1-x))/(a^(1-x)+sqrt(a))`
`=(a^(x))/(a^(x)+sqrt(a))+((a)/(a^(x)))/((a)/(a^(x))+sqrt(a))`
`=(a^(x))/(a^(x)+sqrt(a))+(a)/(a+sqrt(aa)^(x))`
`=(a^(x))/(a^(x)+sqrt(a))+(sqrt(a))/(sqrt(a)+a^(x))=(a^(x)+sqrt(a))/(a^(x)+sqrt(a))=1`
`impliessum_(r=1)^(2n-1)2f((r)/(2n))`
`=2[(f((1)/(2n))+f((2)/(2n))+...+f((n-1)/(2n)+f((n)/(2n))+f((n+1)/(2n))+),(+f((2n-2)/(2n))+f((2n-1)/(2n))]`
`=2[f((1)/(2n))+f((2n-1)/(2n))+f((2)/(2n))+f((2n-2)/(2n))+..+f((n-1)/(2n))+f((n+1)/(2n))+f((n)/(2n))]`
`=2[1+1+1+.....(n-1)` times `+f((n)/(2n))]`
`2[n-1+f((1)/(2))]`
`f((1)/(2))=(a^((1)/(2)))/(a^((1)/(2))+sqrt(a))=(sqrt(a))/(2sqrt(a))=(1)/(2)`
`sum_(r=1)^(n)2f((r)/(2n))=2(n-1+(1)/(2))`
`=2(n-(1)/(2))=2n-1`