Question

If \[\left[ x \right]\] stands for the greatest integer function then the value of \[\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{1}{{1000}}} \right)} \right] + \left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{2}{{1000}}} \right)} \right] + .... + \left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{999}}{{1000}}} \right)} \right]\] is
(1) \[199\]
(2) \[201\]
(3) \[202\]
(4) \[200\]

Answer 1

Hint: In the question, it is given that \[\left[ x \right]\] stands for the greatest integer function which means that it gives the greatest integer less than or equal to the number. To solve this question, we will first divide the given series into two ranges of series i.e., first \[799\] terms different and last \[200\] terms different. Then we can see clearly that the greatest integer value of first \[799\] terms will be \[0\] as they lie between \[0\] and \[1\] so, the terms will vanish individually. And similarly, the greatest integer value of last \[200\] terms will be \[1\] as they lie between \[1\] and \[2\] .Then we just simplify the terms and get the required sum of the given series.

Complete step-by-step answer:
In this question, we are supposed to find the sum of the series \[\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{1}{{1000}}} \right)} \right] + \left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{2}{{1000}}} \right)} \right] + .... + \left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{999}}{{1000}}} \right)} \right]\] with the condition that \[\left[ x \right]\] stands for the greatest integer function.
So, first of all we divide the given series into two ranges of series i.e., first \[799\] terms are different and last \[200\] terms are different.
Therefore, we can write the given series as,
\[\underbrace {\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{1}{{1000}}} \right)} \right] + \left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{2}{{1000}}} \right)} \right] + .... + \left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{799}}{{1000}}} \right)} \right]}_{799terms} + \underbrace {\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{800}}{{1000}}} \right)} \right] + \left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{801}}{{1000}}} \right)} \right] + .... + \left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{999}}{{1000}}} \right)} \right]}_{200terms}\]
Now, let’s first find the values of the first \[799\] terms.
So, \[\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{1}{{1000}}} \right)} \right] = \left[ {\left( {0.2} \right) + \left( {0.001} \right)} \right] = \left[ {0.201} \right]\]
As, we know that the value of greatest integer function is \[0\] when the function lies between \[0\] and \[1\]
Therefore, \[\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{1}{{1000}}} \right)} \right] = 0\]
Similarly, \[\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{2}{{1000}}} \right)} \right] = \left[ {\left( {0.2} \right) + \left( {0.002} \right)} \right] = \left[ {0.202} \right] = 0\]
\[\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{3}{{1000}}} \right)} \right] = \left[ {\left( {0.2} \right) + \left( {0.003} \right)} \right] = \left[ {0.203} \right] = 0\]
.
.
\[\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{799}}{{1000}}} \right)} \right] = \left[ {\left( {0.2} \right) + \left( {0.799} \right)} \right] = \left[ {0.999} \right] = 0\]
Thus, the values of first \[799\] terms is equals to \[0\] .Hence, they get vanish
Now, as we move to the next term the value of greatest integer will be \[1\]
So, let’s check it out.
\[\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{800}}{{1000}}} \right)} \right] = \left[ {\left( {0.2} \right) + \left( {0.8} \right)} \right] = \left[ {1.0} \right]\]
As, we know that the value of greatest integer function is \[1\] when the function lies between \[1\] and \[2\]
Therefore, \[\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{800}}{{1000}}} \right)} \right] = 1\]
Similarly, if we check for all the remaining terms also, we will get the value as \[1\]
i.e., \[\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{801}}{{1000}}} \right)} \right] = \left[ {\left( {0.2} \right) + \left( {0.801} \right)} \right] = \left[ {1.001} \right] = 1\]
\[\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{802}}{{1000}}} \right)} \right] = \left[ {\left( {0.2} \right) + \left( {0.802} \right)} \right] = \left[ {1.002} \right] = 1\]
.
.
\[\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{999}}{{1000}}} \right)} \right] = \left[ {\left( {0.2} \right) + \left( {0.999} \right)} \right] = \left[ {1.199} \right] = 1\]
Thus, from the above calculations, we get
\[\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{1}{{1000}}} \right)} \right] + \left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{2}{{1000}}} \right)} \right] + .... + \left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{799}}{{1000}}} \right)} \right] + \left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{800}}{{1000}}} \right)} \right] + \left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{801}}{{1000}}} \right)} \right] + .... + \left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{999}}{{1000}}} \right)} \right]\]
\[ = \underbrace {0 + 0 + 0 + .. + 0}_{sum{\text{ }}of799terms} + \underbrace {1 + 1 + 1 + ... + 1}_{sum{\text{ }}of200terms}\]
which means the sum of first \[799\] terms will get vanish, therefore we get
\[ \Rightarrow \underbrace {1 + 1 + 1 + ... + 1}_{sum{\text{ }}of200terms}\]
\[ \Rightarrow \left( {200 \times 1} \right)\]
\[ = 200\]
Therefore, the sum of the series \[\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{1}{{1000}}} \right)} \right] + \left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{2}{{1000}}} \right)} \right] + .... + \left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{{999}}{{1000}}} \right)} \right]\] is equal to \[200\]
So, the correct answer is “200”.

Note: While solving this question the most important thing to remember is that,
If \[x \in \left[ {0,1} \right)\] then the value of greatest integer function, \[\left[ x \right] = 0\]
If \[x \in \left[ {1,2} \right)\] then the value of greatest integer function, \[\left[ x \right] = 1\]
Also, we can express the above calculation mathematically as:
\[\left[ {\left( {\dfrac{1}{5}} \right) + \left( {\dfrac{x}{{1000}}} \right)} \right] = \left\{
  0{\text{ }}if{\text{ }}1 < x < 800 \\
  1{\text{ }}if{\text{ }}800 < x < 999 \\
  \right.\]