Question

Let $f\left(n\right)=[\frac{1}{4}+\frac{n}{25}]n$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$, then $\sum _{n=1}^{50}f\left(n\right)$ is equal to

• A. $1399$
• B. $775$
• C. $1433$
• D. $1004$

Answer 1

The correct option is C $1433$

Let $g\left(n\right)=[\frac{1}{4}+\frac{n}{25}]$
Now,
$g\left(n\right)=0$, if $n=1,2,\dots ,18$
$g\left(n\right)=1$, if $n=19,20,\dots ,43$
$g\left(n\right)=2$, if $n=44,45,\dots ,50$
$\therefore \sum f\left(n\right)=0+(19+20+\dots +43)+2(44+45+\dots +50)$
$\sum f\left(n\right)=\frac{25}{2}(19+43)+2\times \frac{7}{2}(44+50)$
$\sum f\left(n\right)=1433$