$10.$ State the formal mathematical definition of $$big$-$oh$"$ notation $($using mathematical symbols$).$

$11.$ Assume that we have an algorithm that prints "Cro" the following number of times. Demonstrate how to find the solution to the runtime of this algorithm

$($hint: use an arithmetic series in the solution$).$

$1+2+3+4+...+$ n

$12.$ Write an algorithm that will print your last name in O$($n$3$ lg n$)$ time.

$13.$ Write an algorithm that will determine the letter and maximum number of duplicate characters for a given string. For example, if given the string "ababac", then you would output $"$a$".$ State the asymptotic complexity of this algorithm using the tightest

bound possible. Remember, do not provide Java syntax as this should be pseudocode.

$14.$ Write the pseudocode for a linear search algorithm and state the asymptotic complexity of this algorithm.

$15.$ Write the pseudocode for the binary search algorithm and state the asymptotic complexity of this algorithm.