Prove that n 5100 theta n100SolutionIn orger to prove n510

Solution

In orger to prove (n+5)^100 = (n^100), we have to prove that (n+5)^100 = O(n^100) and (n+5)^100 = (n^100). Let us first prove the first one:

Let us consider n>5 so 2n>n+5 which implies (2n)^100 > (n+5)^100. So for all n>5 (2^100)(n)^100 > (n+5)^100. So according to definition of big-O (n+5)^100 = O(n^100).

Now let us consider n>0 so n<n+5 which implies (n)^100 < (n+5)^100. So for all n>0 (n)^100 < (n+5)^100. So according to definition of big-O (n+5)^100 = (n^100).

Now that we proved (n+5)^100 = O(n^100) and (n+5)^100 = (n^100), we can conclude that (n+5)^100 = (n^100).