Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009.
Induction
1. Prove that the sum of the first n odd numbers is n2 .
2. Using the product rule
d
(xn ) = nxn−1 .
that dx
d
dx (f (x)g(x))
= f 0 (x)g(x) + f (x)g 0 (x), and the fact that
3. Let an be the sequence defined by a1 =
√
2, an =
√
d
dx x
= 1, use induction to prove
2an−1 .
(a) Show that 1 < an < 2 for all n.
(b) Show that an+1 > an for all n.
NOTE: A sequence like this is called bounded and monotone and is guaranteed to converge.
4. The harmonic numbers are defined by Hn = 1 +
(a) Show that H2n ≥ 1 +
n
2
1
2
+
1
3
+
1
4
+ · · · + n1 .
for all n.
∞
X
1
increases without bound.
(b) Use your answer to (a) to show that
n
n=1
5. Prove that 1 + 23 + 33 + 43 + · · · + n3 =
n(n + 1)
2
2
.
Strong Induction and Well-ordering
1. Prove that if n ≥ 18, then you can make n cents out of just 4- and 7-cent stamps
2. Here we’ll use well-ordering to show that x2 + y 2 = 3xyz has no solutions in positive integers.
(a) Show that any solution must have x ≡ 0 (mod 3), y ≡ 0 (mod 3).
(b) Use well-ordering to show that if there are any solutions, there must be at least one that makes x + y + z
minimal. Call it (x0 , y0 , z0 ).
(c) Combine parts a) and b) to show that ( x30 , y30 , z0 ) must also be a solution.
(d) Explain why this is a contradiction.
Induction-based Puzzles
1. FF I recently paid a visit to a magical island, inhabited by 99 highly intelligent lions and a single goat. While
there, I decided to teach the lions mathematical induction. They seemed fascinated by the subject, and at first
I wasn’t sure what use a lion would have for induction. Then, when my tour ship left the island, the captain’s
first mate told me a story...
He explained that, on that island, the lions eat grass instead of goat, because the goat has magical properties.
Indeed, should a lion eat the magical goat, then over night, that lion would BECOME the magical goat. Until
now, this had seemed to pose a paradox to the lions: if they should eat the goat, then what would stop them
from being eaten the next day? And if none of the lions should eat the goat, then being the goat would be
perfectly safe! But my lecture had solved their conundrum!
To be clear: each lion’s first priority is to survive, and their second is to eat goat rather than grass. Also, except
when the goat is eaten, the lion population remains perfectly stable. And the lions cannot share the meat of one
goat.
When I return to the island, how many lions will I find?
2. F Suppose you host a Ping-Pong tournament with n people in which everyone plays everyone else exactly once
(that is, each person plays n − 1 games). Show that no matter what the outcomes of the matches, you can put
all the participants in a line such that for all i the ith person in line beat the i + 1st person in line.