Question

For $x\in R$, $\left[x\right]$ denotes the greatest integer less than or equal to $x$. The largest natural number $n$ for which
$E=\left[\frac{\pi }{2}\right]+[\frac{1}{100}+\frac{\pi }{2}]+[\frac{2}{100}+\frac{\pi }{2}]...+[\frac{n}{100}+\frac{\pi }{2}]<43$
is

• A. $41$
• B. $42$
• C. $43$
• D. $97$

Answer 1

Answer: (A)

$41$

The value of $\left[\frac{\pi }{2}\right]=1$ , as $1.57<\frac{\pi }{2}<1.58$
So, If we take total $42$ terms, the last term will be $[\frac{41}{100}+\frac{\pi }{2}]$. The term inside is still less than 2. Hence $[\frac{41}{100}+\frac{\pi }{2}]=1$
So the sum of the $42$ terms would be $42$.
If we choose $43$ terms ($n=42$), the total would be $43$.
Therefore maximum value of $n$ is $41$ as $n$ varies from $0$ to $41$.
Answer is option A.