C Roettger, Spring ’16
Name (please print): . . . . . . . . .
Math 201 – Practice Exam 2
Problem 1 If x3 + 3x ≥ 14, prove x ≥ 2.
Problem 2 For n ∈ N, is n3 + n + 1 always a prime? Prove or disprove.
Problem 3 Prove that there are infinitely many primes of the form p =
6n − 1, with n ∈ N. Hint – re-read the proof by contradiction that there are
infinitely many primes, and sort primes according to congruences modulo 6.
Problem 4 Let an be defined by a0 = 0, a1 = 1 and
an+1 = 5an − 6an−1
a) Prove that an+1 > an for all integers n ≥ 0. Don’t spend hours on this see at the end of this practice exam for a hint.
b) Prove that an = 3n − 2n for all integers n ≥ 0.
Problem 5 Let S be a set with n ≥ 1 elements.
a) Prove that S has the same number of subsets of even size as subsets of
odd size.
b) Prove that S has 2n−1 subsets of even size. Note that S itself and ∅ count
as subsets of S.
Problem 6 Prove that an integer is divisible by 24 exactly if it is divisible
by 12 and by 8. Is the same true about being divisible by 4 and by 6?
Problem 7 Let Sn be the L-shape made from 3 squares of size 2n . Prove
that you can tile Sn with copies of S0 (= cover with no gaps or overlapping).
Hint. Re-read the problem about covering a big square with one unit square
taken out.
Hint for problem 4: Prove by strong induction that for all n ≥ 1
0 ≤ 2an ≤ an+1 ≤ 5an .
Or just do part b) first, also by strong induction. It’s not satisfactory because
who on Earth can guess these? but at least once you have these hints,
Induction works.