The range of f(x)=[1+sinx]+[2+sinx2]+[3+sinx3]+...+[n+sinxn]∀x∈[0, π], where [.] where denotes the greatest integer function, is

f(x)=[1+sinx]+[2+sinx2]+[3+sinx3]+−−−+[n+sinxn]for x∈(0, π), sinx, sinx2, sinx3,−−−sinxn ∈(0, 1)if x=0 then f(x)=1+2+3+−−−+n=n(n+1)2if x=π then f(x)=1+[2+1]+3+4−−−+n =(1+2+3+4+−−−+n)+1 =n(n+1)2+1 =n2+n+22Range ={n(n+1)2, n2+n+22}

The range of f(x)=[1+sinx]+[2+sinx2]+[3+sinx3]+...+[n+sinxn]∀x∈[0, π], where [.] where denotes the greatest integer function, is

f(x)=[1+sinx]+[2+sinx2]+[3+sinx3]+−−−+[n+sinxn]for x∈(0, π), sinx, sinx2, sinx3,−−−sinxn ∈(0, 1)if x=0 then f(x)=1+2+3+−−−+n=n(n+1)2if x=π then f(x)=1+[2+1]+3+4−−−+n =(1+2+3+4+−−−+n)+1 =n(n+1)2+1 =n2+n+22Range ={n(n+1)2, n2+n+22}

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$T h e r a n g e o f f (x) = [1 + \sin x] + [2 + \sin \frac{x}{2}] + [3 + \sin \frac{x}{3}] + ... + [n + \sin \frac{x}{n}] \forall x \in [0, π], w h e r e [.]$ where denotes the greatest integer function, is

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${\frac{n^{2} + n - 2}{2}, \frac{n (n + 1)}{2} . \frac{n^{2} + n + 2}{2}}$

${\frac{n^{2} + n - 2}{2}, \frac{n (n + 1)}{2}}$

${\frac{n (n + 1)}{2}}$

${\frac{n (n + 1)}{2}, \frac{n^{2} + n + 2}{2}}$

answer is D.

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Detailed Solution

$\begin{array}{l} f (x) = [1 + \sin x] + [2 + \sin \frac{x}{2}] + [3 + \sin \frac{x}{3}] + - - - + [n + \sin \frac{x}{n}] \\ f o r x \in (0, π), \sin x, \sin \frac{x}{2}, \sin \frac{x}{3}, - - - \sin \frac{x}{n} \in (0, 1) \\ i f x = 0 t h e n f (x) = 1 + 2 + 3 + - - - + n \\ = \frac{n (n + 1)}{2} \\ i f x = π t h e n f (x) = 1 + [2 + 1] + 3 + 4 - - - + n \\ = (1 + 2 + 3 + 4 + - - - + n) + 1 \\ = \frac{n (n + 1)}{2} + 1 \\ = \frac{n^{2} + n + 2}{2} \\ R a n g e = {\frac{n (n + 1)}{2}, \frac{n^{2} + n + 2}{2}} \end{array}$