Let `R=(5 sqrt 5+11)^(2n+1)` and f=R-[R] where [] is the greatest integer function. Prove that Rf=`4^(2n+1)`

If p=(8+3sqrt(7))^n a n df=p-[p],w h e r e[dot] denotes the greatest integer function, then the value of p(1-f) is equal to 1 b. 2 c. 2^n d. 2^(2n)

Let f(n)=[1/2+n/100] , where [.] denotes the greatest integer function,then the value of sum_(n=1)^151 f(n)

Evaluate lim_(n->oo) [sum_(r=1)^n1/2^r] , where [.] denotes the greatest integer function.

If f : R to [-1, 1] , where f(x)=sin(pi/2[x]) (where [.] denotes the greatest integer function), then the range of f(x) is

Let S_n = underset (t = 1) overset n sum r!(n gt 6) , then S_n - 7[S_n/7] (where [.] denotes the greatest integer function) is equal to

Let f : R rarr[-1,oo] and f(x)= ln([|sin 2 x|+|cos 2 x|]) (where[.] is greatest integer function), then -

If f: R rarr R is a function defined by f(x)=[x]cos((2x-1)/2)pi , where [x] denotes the greatest integer function, then f is

Evaluate: int_(-5)^5x^2[x+1/2]dx(w h e r e[dot] denotes the greatest integer function).

Which of the following function is/are periodic? A. f(x)={1,0,x is rationalx is irrational B. f(x)={x−[x],12,2n≤x<2n+12n+1≤x<2n+2 where [.] denotes the greatest integer function n∈Z C. f(x)=(−1)[2xπ, where [.] denotes the greatest integer function D. f(x)=x−[x+3]+tan(πx2), where [.] denotes the greatest integer function, and a is a rational number

lim_(xto0^(-)) (sum_(r=1)^(2n+1)[x^(r)]+(n+1))/(1+[x]+|x|+2x), where ninN and [.] denotes the greatest integer function, equals