Question

For the real number $x$, $\left[ x \right]$ denotes the integral part of x. The value of the following expression is
\[\left[ \dfrac{1}{2} \right]+\left[ \dfrac{1}{2}+\dfrac{1}{100} \right]+\left[ \dfrac{1}{2}+\dfrac{2}{100} \right]+..........+\left[ \dfrac{1}{2}+\dfrac{99}{100} \right]\]
 A. 49
B. 50
C. 48
D. 51

Answer 1

Hint:In the question given, an integral part of $x$, $\left[ x \right]$ that is also known as the step function. The above-given question is an observation-based question that has to be solved on the basis of counting the terms given in the expression, and here it is necessary to find out which term or terms of the expression are exact integers.

Complete step-by-step solution:
The given expression is $\left[ {\dfrac{1}{2}} \right] + \left[ {\dfrac{1}{2} + \dfrac{1}{{100}}} \right] + \left[ {\dfrac{1}{2} + \dfrac{2}{{100}}} \right] + \cdots \cdots + \left[ {\dfrac{1}{2} + \dfrac{{99}}{{100}}} \right]$
That is,
$\left[ {\dfrac{1}{2}} \right] + \left[ {\dfrac{1}{2} + \dfrac{1}{{100}}} \right] + \cdots \cdots + \left[ {\dfrac{1}{2} + \dfrac{{49}}{{100}}} \right] + \left[ {\dfrac{1}{2} + \dfrac{{50}}{{100}}} \right] + \cdots \cdots + \left[ {\dfrac{1}{2} + \dfrac{{98}}{{100}}} \right] + \left[ {\dfrac{1}{2} + \dfrac{{99}}{{100}}} \right]$
In the above expression, $\dfrac{1}{2}$ is the common fraction and the other fraction runs from $\dfrac{1}{{100}}$ to $\dfrac{{99}}{{100}}$ with a difference of$\dfrac{1}{{100}}$. There are a total of 100 terms in the given expression.
So, the term in the above expression becomes an integer only when the other fraction is $\dfrac{1}{2}$ . That is, the other fraction is $\dfrac{50}{100}$.
The term with the fraction $\dfrac{50}{100}$ is the ${{51}^{st}}$ term and its value is 1. From there onwards, the value of every term increases by$\dfrac{1}{{100}}$.
Therefore, the value of the first 50 terms (upto $\left[ \dfrac{1}{2}+\dfrac{49}{100} \right]$) is 0 and the next 50 (upto $\left[ \dfrac{1}{2}+\dfrac{99}{100} \right]$) is 1.
That implies, the value of the given expression,
= $50(0)+50(1)$
Sum = 50
Hence the correct answer is option (B) 50.

Note:To solve the above question, you just need to know what a step function is, and here it is already given in the question as the integral part, which is its definition. These types of questions can only be answered by observation, which comes with practice or an idea. Also, the function is named a step function because its graph looks like a staircase.