# How to proceed with this Sequence question

Let $$a,b$$ be given positive integers such that $$a. Let $$M(a,b)$$ be defined as $$\frac{\sum_{k=a}^b \sqrt{k^2+3k+3}}{b-a+1}.$$

Evaluate $$[M(a,b)]$$ where $$[.]$$ represents greatest integer function.

I have tried to factorise the term inside the square root however I believe it was pretty useless. I have a feeling that it might(?) telescope but I have no clue what to do here. Any help will be appreciated

You should use the estimates $$(x+1)^2. Then you get that $$\frac{\sum_{k=a}^b(k+1)}{b-a+1} Using the fact that $$\sum_{k=0}^nk=n(n+1)/2$$ and the appropriate translations (write $$k=k-a+a$$ so you re-index the sums starting from $$0$$) we get the estimates $$1+\frac{b-a}{2}+a This concludes:
If $$b-a$$ is even, then $$[M(a,b)]=1+(b-a)/2+a$$.
If $$b-a$$ is odd, then $$[M(a,b)]=1/2+(b-a)/2+a$$.