If `f(r)=1+1/2+1/3+1/ra n df(0)=0,t h e nsum_(r=1)^n(2r+1)f(r)` is equal to (a)`nf(n+1)-((n^2-3n+2))/2` (b) `(n+1)^2f(n)-((n^2-3n+2))/2` (c) `(n+1)^2f(n+1)-((n^2+3n+2))/2` (d) `(n+1)^f(n+1)-((n^2-3n+2))/2`

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

The value of sum_(r=1)^n(sum_(p=0)^(r-1) ^nC_r ^rC_p 2^p) is equal to (a) 4^(n)-3^(n)+1 (b) 4^(n)-3^(n)-1 (c) 4^(n)-3^(n)+2 (d) 4^(n)-3^(n)

If x_1, x_2, x_3,...... x _(2n) are in A.P , then sum _(r=1)^(2n) (-1)^(r+1) x_r^2 is equal to (a) (n)/((2n-1))(x _(1)^(2) -x _(2n) ^(2)) (b) (2n)/((2n-1))(x _(1)^(2) -x _(2n) ^(2)) (c) (n)/((n-1))(x _(1)^(2) -x _(2n) ^(2)) (d) (n)/((2n+1))(x _(1)^(2) -x _(2n) ^(2))

The value of sum_(r=0)^n(a+r+a r)(-a)^r is equal to a. (-1)^n[(n+1)a^(n+1)-a] b. (-1)^n(n+1)a^(n+1) c. (-1)^n((n+2)a^(n+1))/2 d. (-1)^n(n a^n)/2

Let f(n)=2cosn xAAn in N , then f(1)f(n+1)-f(n) is equal to (a) f(n+3) (b) f(n+2) (c) f(n+1)f(2) (d) f(n+2)f(2)

If S_n=sum_(r=1)^n(1+2+2^2+ .......+2^r)/(2^r), then S_n is equal to (a) 2^n n-1 (b) 1-1/(2^n) (c) n -1+1/(2^n) (d) 2^n-1

Let f(n)=2cosn xAAn in N , then f(1)f(n+1)-f(n) is equal to f(n+3) (b) f(n+2) f(n+1)f(2) (d) f(n+2)f(2)

Prove that sum_(r = 1)^(n+1) (2^(r +1) C_(r - 1) )/(r (r + 1)) = (3^(n+2) - 2n - 5)/((n+1)(n+2))

lim_(n->oo)1/nsum_(r=1)^(2n)r/(sqrt(n^2+r^2)) equals

f(1)=1, n ge 1 f(n+1)=2f(n)+1 then f(n)=