Discrete Math: Growth of Functions and Big O

Show that $3x^2 + 25$ is on the order of $x^2$ or Big O of $x^2$.

Determine the Big O notation for the following linear time for loop: where the loop prints numbers 0 through n. Analyze its time complexity.

Compute the Big O notation for the given block of code. Assume each statement in the sequence has already been deduced to its Big O.

Given a function named addup, which adds numbers up to a certain number depending on the argument passed, determine the runtime complexity of the function both for the scenario where it iterates with a for-loop and where it computes using the formula sum = $n \times (n + 1) / 2$.

Explain the different time complexities represented by Big O notation, such as $O(1)$, $O(n)$, $O(\log n)$, $O(n \log n)$, and $O(n^2)$, using examples like accessing an element in an array, binary search, looping through elements, and sorting operations.

Analyze the time complexity of a recursive implementation of the Fibonacci sequence.

In 2005, there were a thousand rabbits on an island. The population grows 8% every year. At this rate, how many rabbits will there be on the island by 2020?

How long will it take for the sample to contain 500 million counts of bacteria, given it triples every 15 minutes?

For a given function $f(n) = 2n + 3$, determine the Big O, Omega, and Theta notation expressions.

Prove that a specific algorithm has an upper bound according to Big-O notation.