Problem 106
Question
If \(\\{x\\}\) and \([x]\) represent fractional and integral part of \(x\), then the value of \([x]+\sum_{r=1}^{2000} \frac{\\{x+r\\}}{2000}\) is (A) \(x\) (B) \(2000 x\) (C) 0 (D) None of these
Step-by-Step Solution
Verified Answer
The value is (A) \(x\).
1Step 1: Understand the Notations
The notation \( [x] \) denotes the greatest integer less than or equal to \( x \), which is called the "floor" of \( x \), and \( \{x\} \) represents the fractional part of \( x \), defined as \( \{x\} = x - [x] \).
2Step 2: Analyze the Series
The series in the problem is \( \sum_{r=1}^{2000} \frac{\{x + r\}}{2000} \). Each term \( \{x + r\} \) is the fractional part of the number \( x + r \) which equals to \( \{x + r\} = x - [x] + r - [r] \). However, since \( r \) is an integer, \( [r] = r \), the expression simplifies to \( \{x + r\} = \{x\} \). Therefore, each term in the sum is just \( \frac{\{x\}}{2000} \).
3Step 3: Evaluate the Sum
Now we evaluate the sum: \( \sum_{r=1}^{2000} \frac{\{x + r\}}{2000} = \frac{1}{2000} \sum_{r=1}^{2000} \{x\} = \frac{1}{2000} \times 2000 \times \{x\} = \{x\} \).
4Step 4: Combine with the Floor
Combine \([x]\) with the sum: \([x] + \sum_{r=1}^{2000} \frac{\{x + r\}}{2000} = [x] + \{x\}\). By the properties of fractional and integral parts, \([x] + \{x\} = x\).
5Step 5: Conclusion: Identify the Correct Answer
After evaluating the expression, the correct answer is \(x\), which matches choice (A).
Key Concepts
Floor FunctionSeries EvaluationGreatest Integer Function
Floor Function
The floor function is a mathematical concept that rounds a real number down to the nearest integer. It's often represented by the notation \([x]\), signifying the greatest integer less than or equal to \(x\). This concept is particularly useful when you want to strip a number of its fractional component.
For example, if you have \(x = 3.7\), the floor of \(x\), denoted as \([3.7]\), is 3. Similarly, for a negative number, such as \(-2.3\), \([-2.3]\) equals \(-3\), because \(-3\) is the greatest integer less than or equal to \(-2.3\).
For example, if you have \(x = 3.7\), the floor of \(x\), denoted as \([3.7]\), is 3. Similarly, for a negative number, such as \(-2.3\), \([-2.3]\) equals \(-3\), because \(-3\) is the greatest integer less than or equal to \(-2.3\).
- The floor function always results in an integer value.
- It can be visualized as the largest step downward from a number on a number line.
- Even whole numbers are unaffected by this operation, e.g., \([5] = 5\).
Series Evaluation
Series evaluation is a process of finding the value of a series, which is a sum of terms. It involves carefully analyzing each element of the series to determine how they collectively impact the total sum.
In the given exercise, the series \(\sum_{r=1}^{2000} \frac{\{x + r\}}{2000}\) involves evaluating summands that include the fractional parts of numbers. The fractional part, \(\{x\}\), of a number \(x\) is the portion of \(x\) beyond its integer component, calculated as \(x - [x]\) where \([x]\) is the floor of \(x\).
In the given exercise, the series \(\sum_{r=1}^{2000} \frac{\{x + r\}}{2000}\) involves evaluating summands that include the fractional parts of numbers. The fractional part, \(\{x\}\), of a number \(x\) is the portion of \(x\) beyond its integer component, calculated as \(x - [x]\) where \([x]\) is the floor of \(x\).
- Each term involves dividing the fractional part of \(x + r\) by 2000.
- The constant structure of \(\{x + r\} = \{x\}\) simplifies calculations.
- The end result is effectively scaled by the number of terms, simplifying to \(\{x\}\).
Greatest Integer Function
The greatest integer function, synonymous with the floor function, refers to the method of finding the greatest integer that is less than or equal to a given real number. This understanding is essential when dissecting expressions involving both integral and fractional parts.
In mathematical terms, the greatest integer function \([x]\) for a number \(x\) is intuitive: imagine any decimal number and its nearest lower whole number. This integer plays a crucial role in many calculations and simplifies the integration of numbers with fractional parts.
In mathematical terms, the greatest integer function \([x]\) for a number \(x\) is intuitive: imagine any decimal number and its nearest lower whole number. This integer plays a crucial role in many calculations and simplifies the integration of numbers with fractional parts.
- It ensures that any real number is expressed clearly in terms of its integer and non-integer components.
- Provides a structured way to assess how numbers interact in summations or sequences.
- Essential in problems involving floor calculations, allowing streamlined computations.
Other exercises in this chapter
Problem 104
The range of the function \(y=\frac{x-[x]}{1-[x]+x}\) (A) \(\left(0, \frac{1}{2}\right)\) (B) \(\left[0, \frac{1}{2}\right]\) (C) \(\left[0, \frac{1}{2}\right)\
View solution Problem 105
If the domain for \(y=f(x)\) is \([-3,2]\), then the domain of \(g(x)=f\\{|[x]|\\}\) is (A) \((-2,3)\) (B) \([-2,3]\) (C) \([-2,3)\) (D) \((-2,3]\)
View solution Problem 107
If \(f:(0, \pi) \rightarrow R\) be defined by \(f(x)=\sum_{k=1}^{n}([1+\sin k x])\), where \([x]\) denotes the integral part of \(x\), then the range of \(f(x)\
View solution Problem 108
Consider a function \(f(n)\) defined for all \(n \in N\). The function satisfies the following two conditions (i) \(f(1)+f(2)+f(3)+\ldots\) to \(\infty=1\) (ii)
View solution