Question

If $\sum _{n=1}^k[\frac{1}{3}+\frac{n}{90}]=21$ where $\left[x\right]$ denotes the integral part of $x$, then $K$$=$

• A. $84$
• B. $80$
• C. $85$
• D. None of these

Answer 1

The correct option is B $80$

From equation we can say that when $\frac{n}{90}=\frac{2}{3}$ ,
$[\frac{1}{3}+\frac{n}{90}]=1$
After that for each value of $n$ till $\frac{n}{90}<\frac{5}{3}$ we have $[\frac{1}{3}+\frac{n}{90}]=1$
So the value of $n$ can vary from $60$ to $80$. As $n=80$ will be the 21st term of the series and hence the sum will be $21$ at $n=80$.
So option B is correct.