Hello I need help with this part of the assignment , thank you

Problem(7) (3 points) Derive the solution(s) to the equation ar2 4 br + c = 0, a # 0 is that (the quadratic formula) r = =byblac 2a Problem(8) Prove the following two famous Identity/Principle in Discrete Mathernatics (a) (5 points) (Pascal's Identity) Let n and k be positive integers with n 2 k, show that ("*" ) - ( .". ) . (") (b) (5 points) (The Generalized Pigeonhole Principle) Show that if N objects are placed into & boxes, then there is at least one box containing at least [ ] objects. (b') (2 points) Among 45 students in our Discrete Math at least how many were born in the same month? Problem(9) (7 points) The Fibonacci numbers, fo, fi, f2, , are defined by the (recursive) equations fo =0, f1 =1, and fk = fk-1 + fk-2 for k = 2, 3, 4, ... Use mathematical induction to prove that n = 0, 1, 2, 3, ... Problem(10) The Fibonacci numbers, fo, fi, f2, .., are defined by the (recursive) equa- tions fo =0, fi = 1, and fa = fu-l+ fu-2 for n = 2, 3, 4, .. .. (a) (5 points) Show that futifu-1 - fa = (-1)" when n is a positive integer. (b) (5 points) Prove that fo - fit fa - fat . - fan-i + fan = fan-1 - 1 when n is a positive integer. Problem(11) Theorem 1 on page 542 states the following: Let c, and c2 be real numbers. Suppose that 7 - cir - c2 - 0 has two distinct roots r, and r2. Then the sequence {an } is a solution of the recurrence relation an = C10-1 + C2dn-2 if and only if an = ort + car? for n = 0, 1, 2, . . ., where a and a2 are constants. (a) (5 points) Verify that an = cift + or, satisfies the recurrence relation an = Cid,-1 + C2an-2- (b) (3 points) What is the solution of the recurrence relation an - 7an-1 - 10an-2 for n 2 2, do = 2, a1 = 1. (c) (5 points) Prove that fo - a-g "-",Vn c N, where a and 3 are the two roots of r-r-1 = 0. [Hint: Use mathematical induction and the fact that a = a + 1 and B' = 3 + 1.] Problem(12) (7 points) Consider n independent repetitions of the simple success-failure experiment (with success probability p). Let X be the random variable that denotes the number of successes in the n trials. (a) What are the values that the random variable X takes? (b) Write down the expression (the density function of the binomial distribution) P(X = k). (c) Show that Ek_, P(X - k) = 1? (d) (4 points) Show that E[X] = np