Question

[6 marks] Arrange the functions (1.5)n , n100, log n, n!, and n99 + n98 in a list so that each is a big-O of the next.

Ans: log n, (n99+n98), n100, (1.5)n , n!

Answer 1

There is a typing mistake in your question, I oberved the function from given answer, So please correct in commnet if my obervation is wrong. I will edit my answer or try to help you in comment.
Question: Arrange the functions ($(1.5)^n , n^{100}, \log n, n!$, and
99 98 n +n
in a list so that each is a big-O of the next.
Solution:


15
is an exponential function with base 1.5
$n^{100}$ is a polynomial function of degree 100
log n is a logarithmic function
is a factorial funtion

99 98 n +n
is polynomial function.

The increasing order of Big O notation is as given below:

Constant functions Logarithmic functions Polynomial functions in increasing order of degree) Exponential function (in increasing order of the base) Factorial functions


There is no constant function in our question. Now comes on logarthmic funnction
log n
Next is polynomial function $n^{100}$ ( Degree 100 )  and
99 98 n +n
(Degree 99)
log n <
99 98 n +n
< $n^{100}$
Exponential function
15

log n <
99 98 n +n
< $n^{100}$ <  
15

And the final one is Factorial function
So, big O notaion in increasing order is given by:
log n <
99 98 n +n
< $n^{100}$ <  
15
<