Math Is Fun Forum

How can I sum the following sequence:

∑ k∈{⌊n∕2⌋…n-1}k = ⌊n∕2⌋ + ⌊n+1∕2⌋ + ⌊n+2∕2⌋ + ...... + (n-1)

(In another form)

n-1
  ∑    k  =  ⌊n∕2⌋ + ⌊n+1∕2⌋ + ⌊n+2∕2⌋ + ...... + (n-1)
⌊n∕2⌋

What I think is discard the floor and sum what inside each floor !! This is just a guess.

Give me any hint or general formula that helps me to sum them

Thanks

Hi

I need to prove the following statements:

1) for any a>1, and any b, a^n ∈ ω(n^b).

We have to prove f(n) > C.g(n) for all C>0, n0>0 and n>=n0
a^n > C . n^b

Since a > 1 , I think a >= n and a > b
which assures that the left hand side will be always greater than the right hand side.

Is that right ?

2) We have f(n)= n^2 and g(n)=42.
Is f(n) ∈ O (g(n)) or f(n) ∈ Ω(g(n)) ?

What I think is f(n) ∈ Ω(g(n))
n^2 >= C.g(n)
n must be >= 7 and C = 1
Is this prove right or not ?

Thanks

Hi

I have the following statement

2^(⌊lg n⌋+⌈lg n⌉)∕n ∈ Θ(n)

I need to prove it. The first thing that I am going to do is to expand the exponential .. someone gave me a hint about expanding the exponential but I didn't understand it.

Can someone explain that to me plz ?

Thanks