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1. lg√n ∈ O(lg n) (√n means square root of n)

2. lg n ∈ O(lg√n)

3. 2n+1 ∈ O(2n)

1. (ab)n = an bn for every natural number n

2. 13 +23+33+…+n3 = (1+2+3+…+n)2

3. 1+3+5+7+…+(2n-1) = n2

(note that log2n means (log n)2.)

2. Suppose your name was George Gershwin. Write a complete main method that would print your last name, followed by a comma, followed by a space and your first name.

3. Write an expression that attempts to read a double value from the TextBox textBox and stores it in a Double variable, x, that has already been declared.

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Hint: Keep track of the nesting depth of brackets. Initially the depth is 0. The depth increases with each opening bracket and decreases with each closing bracket. What are the constraints on the value of the nesting depth for all brackets to be matched?

Here are some examples to show how your function should work.

>>> matched("zb%78")

True

>>> matched("(7)(a")

False

>>> matched("a)*(?")

False

>>> matched("((jkl)78(A)&l(8(dd(FJI:),):)?)")

True

Integer division is defined as whole-number division. For example, 1/2 (as integers) = 0. Whereas, if we specified 1/2 (as "floats"; i.e., floating-point numbers), the result is 0.5. In C++, the result of an arithmetic operation (e.g., whether the computer computes the result as an integer or as a float) is specified by the data type.

Your task is to make edits within main such that, when run, the program produces the following console output:

Enter two integers: [user-specified value for a] [user-specified value for b]

Integer division: [a]/[b] = [a]/[b]

Floating-point division: [c]/[d] = [c]/[d]

Mixed-type division: [a]/[d] = [a]/[d]

Mixed-type division: [c]/[b] = [c]/[b]