Task:

Direction: Prove the following Statements. 1. Let a and b be integers, not both 0, where gcd(a,, b) = d. Prove that for any positive integer k, gchca, lab) 2 led. Hint: Use the following results: a. Let m, n he integers. An integer s is the greatest common divisor of m and \"n. whenever it satises the following i. s | m and s | 71. ii. If I is an integer which divides m and \"n,? then I divides s. I). The greatest common divisor s can be written as a linear combi nation of m. and n; that is s=mf+ng for some integers f and g. 2. Prove that :2". - i! = (n + 1)! 1 for every positive integer n. i=1 3. A sequence is dened recursively by (11 = 1, a2 = 4, 013 = 9 and an = an_1 a.n_2 + an_3 + 2(21'2. 3) for \"n, 2 4. Conjecture a formula for an and prove that your conjecture is correct