# Math 2200-1. Practice Quiz 2. Fall 2008. ```Math 2200-1. Practice Quiz 2. Fall 2008.
Name:
November 13, 2008
Problem 1:
Problem 2:
Problem 3:
Problem 4:
Problem 5:
Problem 6:
Problem 7:
(There will be only six problems in the actual exam!)
Total:
/100
Instructions: The exam is closed book, closed notes and calculators are not allowed. You
are only allowed one letter-size sheet of paper with anything on it.
You will have 75 minutes for this test. The point value of each problem is written next to the
problem - use your time wisely. Please show all work, unless instructed otherwise. Partial credit
will be given only for work shown.
1
2
Problem 1. The sequence of Lucas numbers is defined recursively by ℓ0 = 2, ℓ1 = 1, and
ℓn = ℓn−1 + ℓn−2 , for n ≥ 2.
(1) Find ℓ8 .
(2) Show by induction that
ℓ20 + ℓ21 + &middot; &middot; &middot; + ℓ2n = ℓn ℓn+1 + 2,
whenever n is a nonnegative integer.
Problem 2. Prove by induction that 6 divides n3 − n whenever n is a nonnegative integer.
Problem 3. For which nonnegative integers n is n! ≥ n2 ? Prove your answer by induction.
Problem 4. Given a set of any 15 integers, show that one can always choose two of them so
that their difference is a multiple of 14.
Problem 5. How many permutations of the 26 letters of the English alphabet do not contain
any of the strings fish, rat, or bird.
Problem 6. A coin is flipped 10 times where each flip comes up either heads or tails. How
many possible outcomes
(a) are there total?