2. (8pts) Answer the five questions regarding the sum n O f(n) = for n E Z+. Allowed key strokes: power '^', lg lg', \lfloor', ' floor', [\lceil', 1 ceil' O So type \lceil \lg k ceil to express [lg k] You may not use all the key strokes. [lg k] C. a. (1pt) Find the value of f(8) b. (2pts) Find the value of f(21) for any given j E Z+ in a closed form. Enter your answer in the form of (.......) 2^(.....) + 1 completing the two missing parts ....... Enter the whole expression in this form. (1pt) Given n E Z+, what is the largest value of j such that 2 n? In 2-d and 2-e, we find f(n) for any n in a closed form including just n. The answer can be written as f(n) = X - 2Y Both X and Y are expressions with just one variable n. d. (3pts) Find X. It must be shortest possible with the largest term in the first position and the constant in the last. No parentheses should be included. e. (1pts) Find Y of the minimum length. If any, a constant must be placed at the end.