Find the range of $f(x) = [1+\sin x]+\left[2+\sin\frac{x}{2}\right]+\left[3+\sin\frac{x}{2}\right]+...+\left[n+\sin\frac{x}{n}\right]\, ∀\,x∈[0,π],n∈N$ ([.] denotes greatest integer function).
Answer & explanation
Correct answer: option 3
$f(x) = [1+\sin x]+\left[2+\sin\frac{x}{2}\right]+\left[3+\sin\frac{x}{3}\right]+...+\left[n+\sin\frac{x}{n}\right]$
for, $x∈(0,π)$
$\sin x, \sin \frac{x}{2},\sin\frac{x}{3}, \sin\frac{x}{4}, \sin\frac{x}{5},....\sin\frac{x}{n}∈(0,1)$
∴ for $x ∈ (0, π)$
$f(x)=1+2+3+....+n=\frac{n(n+1)}{2}$
For $x = 0, f(x)=1+2+3+....+n=\frac{n(n+1)}{2}$
for $x = π, f(x)=1+[2+1]+3+4+....+n$
$=(1+2+3+...+n)+1$
$=\frac{n(n+1)}{2}+1=\frac{n^2+n+2}{2}$
∴ Range is $\left\{\begin{matrix}\frac{n(n+1)}{2},\frac{n^2+n+2}{2}\end{matrix}\right\}$