Question

Arrange the functions (1.5)ⁿ,n¹⁰⁰,(log n)³,10ⁿ,(n!)², and (n⁹⁹+n⁹⁸) in a list so that each function is big-O of the next function.

Answer 1

To arrange the functions in a list based on their big-O relationship, we need to determine the growth rate of each function as n approaches infinity. After analysis, the list is: (1.5)ⁿ, 10ⁿ, (n⁹⁹+n⁹⁸), n¹⁰⁰, (n!)², (log n)³.

Step-by-step explanation:

To arrange the functions in a list based on their big-O relationship, we need to determine the growth rate of each function as n approaches infinity. Let's analyze each function:

1. (1.5)ⁿ: This is an exponential function that grows exponentially as n increases. It is of higher order than all the other functions.
2. n¹⁰⁰: This is a polynomial function with a constant exponent. It grows at a slower rate than exponential functions, but faster than logarithmic functions.
3. (log n)³: This is a logarithmic function with a cube exponent. It grows at a slower rate than polynomial or exponential functions.
4. 10ⁿ: This is an exponential function with a base of 10. It grows at a faster rate than polynomial, logarithmic, and factorial functions.
5. (n!)²: This is a factorial function with a square exponent. It grows at a slower rate than exponential functions.
6. (n⁹⁹+n⁹⁸): This is a polynomial function with two terms. It grows at a slower rate than exponential functions, but faster than logarithmic functions.

Based on this analysis, we can arrange the functions in the following list: