Let `f(n)=2cosn xAAn in N ,` then `f(1)f(n+1)-f(n)` is equal to`

Let a,b, c in R. " If " f(x)=ax^(2)+bx+c is such that a+b+c=3 and f(x+y)=f(x)+f(y)+xy, AA x,y in R, " then " sum_(n=1)^(10) f(n) is equal to

Let f(n)=1+1/2+1/3++1/ndot Then f(1)+f(2)+f(3)++f(n) is equal to (a) nf(n)-1 (b) (n+1)f(n)-n (c) (n+1)f(n)+n (d) nf(n)+n

If x in{1,2,3......9} and f_n(x) =x x x ...x (n digits ), then f_n^2(3)+f_n(2) is equal to

Let f : R to R be a real function. The function f is double differentiable. If there exists ninN and p in R such that lim_(x to oo)x^(n)f(x)=p and there exists lim_(x to oo)x^(n+1)f(x) , then lim_(x to oo)x^(n+1)f'(x) is equal to

Let f(1)=1 and f(n)=2overset(n-1)underset(r=1)sum f( r ) . Then overset(m) underset(n=1)sum f(n) is equal to

Let f be a function such that f(x + y) = f(x) + f(y) forall x , y in R , if f(1) = k, then show that f(n) is equal to nk.

Let f(x)=(x-1)^4(x-2)^n ,n in Ndot Then f(x) has (a) a maximum at x=1 if n is odd (b) a maximum at x=1 if n is even (c) a minimum at x=1 if n is even (d) a minima at x=2 if n is even

Let f(x)=x^2-2x ,x in R ,a n dg(x)=f(f(x)-1)+f(5-f(x))dot Show that g(x)geq0AAx in Rdot

Let f(x)=x+f(x-1) for AAx in R . If f(0)=1,f i n d \ f(100) .

If f and g are two functions defined on N , such that f(n)= {{:(2n-1if n is even), (2n+2 if n is odd):} and g(n)=f(n)+f(n+1) Then range of g is (A) {m in N : m= multiple of 4 } (B) { set of even natural numbers } (C) {m in N : m=4k+3,k is a natural number (D) {m in N : m= multiple of 3 or multiple of 4 }