STOR 215 Sample Questions for Midterm 2 Here are some sample questions for the second midterm exam Questions on the actual midterm will be di erent Remember that all exams in the course are closed book and closed notes and calculators are not permitted Most questions will have multiple parts and in many cases di erent parts of the same question are unrelated so if you cannot solve one part of a question you should still try the other parts 1 Find a closed form expression for the sequence de ned by the recurrence relations a0 3 and an 1 an 2 for n 0 2 Show that if a b mod m and c d mod m then ac bd mod m 3 Give a simple numerical expression for cid 80 n cid 1 cid 0 n k k 0 4 In each of the following cases indicate whether the set described is nite or in nite You need not prove anything a The rational numbers between 1 and 3 b The real numbers between 1 and 3 b The set of real numbers x such that 12x 4 5 c The set of real numbers 1 n n 1 5 Audrey has a large jar lled with dimes nickels and quarters In how many ways can she leave home with 10 coins from the jar under the following conditions a There are no constraints b She takes at least 2 dimes 2 nickels and 3 quarters 6 Find the numerical value of a cid 80 3 b cid 80 8 c cid 80 3 n cid 99 n 1 cid 98 cid 80 2 j 1 1 i j j 1 j i 1 7 Carefully describe the setting and procedure for proof by weak induction 8 A company has 850 employees who were born in the United States What can you say about the following a The number of employees born in the same state b The number of employees born in Arizona c The number of employees sharing the same last two digits of their Social Security number 9 A board game ends with a win W tie T or loss L Suppose that you play the game 6 times in a row Let an outcome be the record of your wins losses and ties over the 6 plays for example WLLTWL or LLTWTT a How many possible outcomes are there for the 6 plays b How many outcomes begin with WWL c How many outcomes begin with two wins and end with two losses d How many outcomes begin with two wins or end with a loss 10 Find the following a The prime factorization of 390 b gcd 55 231 c 53 mod 6 d 87 mod 11 11 A high school athlete heading to college wishes to give away 20 di erent trophies to 3 family members If she intends to give 5 trophies to the rst family member 5 to the second and 10 to the third how many ways can she give the trophies away