# Solving Fractional Part Sum S(n): a,b,n Natural Non-Null Numbers

• redount2k9
In summary, the conversation involved a request for help in solving a mathematical problem involving a series of fractions and the use of the fractional part and greatest common divisor. The problem was eventually solved with the help of a math expert on another forum.
redount2k9
Hi everyone!
How to solve this: S(n) = { (a+b)/n } + { (2a+b)/n } + { (3a+b)/n } + ... + { (na+b)/n } where {x} represents fractional part of x. a,b,n are natural non-null numbers and (a,n)=1.

I dont need only an answer, i need a good solution.

Thanks!

Welcome to PF;
Have I understood you...
$$S(n)=\frac{a+b}{n}+\frac{2a+b}{n}+\cdots +\frac{(n-1)a+b}{n}+\frac{na+b}{n}$$ ... ... if this is what you intended, then it looks straight forward to simplify: notice that each term is over a common denominator ... you should be able to see what to do from there.

Note: I don't know what you mean by "{x} is the fractional part of x" or "(a,n)=1".

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redount2k9 said:
You didn't understand.
We say that a number x = {x} + [x]
http://en.wikipedia.org/wiki/Fractional_part
(a,n) = 1 that means the greatest common factor
Ex: (2, 3)=1
(23, 29)=1
(14, 19)=1
Hope this helps but I think
you know physics better because this notions are learned in middle school.
Thanks!
Thanks for the detailed description.
You did provide the context with "fractional part" but I didn't get it because this was not taught that way, with those words, in NZ when I went to "middle school" (though I may have missed that class due to dodging dinosaurs and contemplating the possibilities of this new-fangled "wheel" thingy.)

Mind you - (a,b) for "greatest common divisor" (factor - whatever) would be an older and ifaik uncommon notation - it is more usual to see "gcd(a,b)" instead. This sort of thing makes international forums more fun :D

So...
$$\sum_{i=1}^n \left \{ \frac{ia+b}{n} \right \}=\sum_{i=1}^n \frac{ia+b}{n}-\sum_{i=1}^n \left \lfloor \frac{ia+b}{n} \right \rfloor$$

It occurs to me that the properties of the floor function may help here?
...

Note: This is the source of the problem http://www.viitoriolimpici.ro/uploads/attach_data/112/45/26//4e02c08p03.pdf[/QUOTE]

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Thanks anyway man, a very good math guy helped me on other forum with a modulo n solve. It was the best Ive ever seen. It is solved now, but thanks!

Cool - link to the solution?

Thanks :)

## 1. What is the purpose of solving fractional part sum S(n)?

The purpose of solving fractional part sum S(n) is to find the sum of the fractional parts of a sequence of numbers. This can be helpful in various mathematical and scientific applications, such as calculating probabilities or analyzing patterns in data.

## 2. How do you solve for S(n)?

To solve for S(n), you need to first determine the formula for the fractional part sum. This can be done by breaking down the sequence into individual fractions and then using algebraic manipulation to simplify the expression. Once you have the formula, you can plug in the values of a, b, and n to get the final answer.

## 3. Can S(n) be negative?

Yes, S(n) can be negative. This is because the fractional part of a number can be negative if the integer part is negative. For example, if the number is -2.5, the fractional part would be -0.5. Therefore, the sum of the fractional parts of a sequence of negative numbers can result in a negative value for S(n).

## 4. What are the restrictions on the values of a, b, and n?

The values of a, b, and n must be natural (positive whole) numbers and cannot be equal to 0. This is because the fractional part of a number only exists for non-integer values. Additionally, n cannot be a null (0) number as it is used as the upper limit of the sequence.

## 5. How is solving fractional part sum S(n) useful in real-world applications?

Solving fractional part sum S(n) can be useful in various real-world applications, such as finance, statistics, and data analysis. For example, in finance, it can be used to calculate the expected returns on investments. In statistics, it can be used to determine the probability of a certain event occurring. In data analysis, it can be used to identify patterns and trends in a dataset.

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