Evaluating Hermite Identity with Integers: Help with Homework Equations

[rattanjot14]

Homework Statement

Let a and b be integers and m an integer >1 Evaluate

[b/m] + [(b+a)/m]+ [(b+2a)/m]+ [(b+3a)/m]+ [(b+4a)/m]+ [(b+5a)/m]+...+ [(b+(m-1)a)/m]

Homework Equations

The Attempt at a Solution

i tried to use hermite identity.

[x] + [x + 1/n] + [x + 2/n] +...+ [x + (n-1)/n] = [nx]

assuming x = b/m and 1/n = a/m.. but a/m is not an integer so i can't use it. I m stuck what to do?

 

[lurflurf]

Funny you should mention that, just last week I read about it in pp 90-94 of Concrete Mathematics by Ronald L. Graham , Donald E. Knuth, and Oren Patashnik. Several interesting things are the dependence of the result on gcd(a,m) ,the fact that

$$\sum_{0 \le k < m} \left[ \frac{b+k \, a}{m}\right] = \sum_{0 \le k < a} \left[ \frac{b+k \, m}{a}\right], \, \, \, \, \, \, \, \, \, \mathop{integers \, \, \, a,m>0} $$

the closely related integrals

$$\frac{1}{m} \int_0^m (a \, x+b) \! \mathop{dx}=\frac{1}{a} \int_0^a (m \, x+b) \! \mathop{dx}=\frac{a \, m}{2}+b$$

The book uses several special cases to deduce the general one. Give it another try. If you have trouble describe the methods you tried and those you have know.