Use the definition of Θ -notation to show that ⌈ n + 1 2 ⌉ is Θ ( n ) .

To show:

That ⌈n+12⌉is Θ(n) using the definition of Θ notation.

Given information:

Let f(n)=⌈n+12⌉ and g(n)=n, for any integer n≥1.

Formula used:

Let f and g be real valued functions defined on the same nonnegative integers, with g(n)≥0 for every integer n≥r, where r is positive real number.

Then,

f is of order g, written f(n) is Θ(g(n)), if and only if, there exist positive real numbers A,B and k≥r such that

Ag(n)≤f(n)≤Bg(n) for every integer n≥k.

Proof:

Let f(n)=⌈n+12⌉ and g(n)=n, for any integer n≥1.

Hence,

For all n≥0, n+12>0 and n≥1

Then, ⌈n+12⌉≥0 and n≥1

Obviously, f(n)≥0 and g(n)≥0 

Suppose A=1 and B=