If {x} and [x] represent fractional and integral part of x respectively, then find the value of

step 1 As we know that fractional part of x plus r equal fractional part of x as x belongs to integer.

If {x} and [x] represent fractional and integral part of x...

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Key Concepts

Periodicity under Integer Addition

A key concept is the fact that when an integer is added to a real number, the fractional part remains unchanged. This periodic property simplifies analysis in problems involving translations of numbers by integers and is crucial in dealing with sums where the fractional part function is applied after adding integer values.

Summation of Constant Terms

When summing over a range in which each term is identical, the sum can be simplified by multiplying the constant term by the number of terms. This concept is widely applicable in series and summation problems and is instrumental in reducing complex expressions to simpler forms.

Integral Part Function

This function returns the greatest integer less than or equal to a given number. It essentially removes the fractional component, leaving only the whole number part. This concept is fundamental in various areas of arithmetic and number theory, where the separation of the integer and fractional parts is necessary for analysis.

Fractional Part Function

The fractional part function gives the difference between a number and its integral part, effectively isolating the portion of the number that lies after the decimal point. It is used to analyze periodic behavior and is important in understanding the properties of real numbers in many contexts.

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