Home
Class 12
MATHS
If f(x)=(a^x)/(a^x+sqrt(a ,)),(a >0), th...

If `f(x)=(a^x)/(a^x+sqrt(a ,)),(a >0),` then find the value of `sum_(r=1)^(2n1)2f(r/(2n))`

Text Solution

Verified by Experts

`f(x)=(a^(x))/(a^(x)+sqrt(a))`
or `f(1-x)=(a^(1-x))/(a^(1-x)+sqrt(a))`
`=(a^(1))/(a^(1)+sqrt(a)a^(x))`
`=(sqrt(a))/(sqrt(a)+a^(x))`
` or f(x)+f(1-x)=1`
` :. f((1)/(2))=(1)/(2)`
so, `sum_(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-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((1)/(2))}`
`=2{[f((1)/(2n))+f(1-(1)/(2n))]+[f((2)/(2n))+f(1-(2)/(2n))]+ ... +[f((n-1)/(2n))+f(1-(n-1)/(2n))]+(1)/(2)}`
`=2[1+1+1+...+(n-1)" times "]+1`
`=2n-1`
Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 1.1|15 Videos

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 1.2|5 Videos

  • RELATIONS AND FUNCTIONS

    CENGAGE ENGLISH|Exercise Archives(Matrix Match Type)|1 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE ENGLISH|Exercise Archives (Numerical Value Type)|3 Videos

  • SCALER TRIPLE PRODUCTS

    CENGAGE ENGLISH|Exercise DPP 2.3|11 Videos

Similar Questions

Explore conceptually related problems

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)

If n in N such that is not a multiple of 3 and (1+x+x^(2))^(n) = sum_(r=0)^(2n) a_(r ). X^(r ) , then find the value of sum_(r=0)^(n) (-1)^(r ).a_(r).""^(n)C_(r ) .

If 2f(xy) =(f(x))^(y) + (f(y))^(x) for all x, y in R and f(1) =3 , then the value of sum_(t=1)^(10) f(r) is equal to

Suppose f and g are two functions such that f,g: R -> R f(x)=ln(1+sqrt(1+x^2)) and g(x)=ln(x+sqrt(1+x^2)) then find the value of xe^(g(x))f(1/x)+g'(x) at x=1

If the function f: R- {0}-> R given by f(x)=1/x-2/(e^(2x)-1) is continuous at x=0, then find the value of f(0)

If (4x^(2) + 1)^(n) = sum_(r=0)^(n)a_(r)(1+x^(2))^(n-r)x^(2r) , then the value of sum_(r=0)^(n)a_(r) is

If f(x+y+z)=f(x)+f(y)+f(z) with f(1)=1 and f(2)=2 and x,y, z epsilonR the value of lim_(xtooo)sum_(r=1)^(n)((4r)f(3r))/(n^(3)) is ……….

If C_(r) be the coefficients of x^(r) in (1 + x)^(n) , then the value of sum_(r=0)^(n) (r + 1)^(2) C_(r) , is

f(x)+f(1-1/x)=1+x for x in R-{0,1}. Find the value of 4f(2).

If f(x)=(4+x)^(n),"n" epsilonN and f^(r)(0) represents the r^(th) derivative of f(x) at x=0 , then the value of sum_(r=0)^(oo)((f^(r)(0)))/(r!) is equal to