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Transcript
HOMEWORK 1: DUE THURSDAY SEPTEMBER 29
INSTRUCTOR: PAUL APISA
For each problem, communicating how you reasoned your way to the answer you’re
submitting is far more important than the answer itself. “Show” is shorthand for “Write a
proof or produce a counterexample”.
1. Practice with the Rational Numbers
Remember that a number r is said to be rational if r = ab where a and b are integers
and b is nonzero. Recall that the integers are the counting numbers along with 0 and their
negatives, i.e. {. . . , −2, −1, 0, 1, 2, . . .}.
Exercise 1: If a and b are irrational numbers is a + b irrational? What about ab?
√
√
Exercise 2: Imitate the proof of the irrationality of 2 to show that 3 is irrational.
2. Practice with Induction and Combinatorics
Recall that
is the number
Pof ways to choose k objects from a collection of n objects.
Recall too that, by definition, nk=0 ak = a0 + a1 + . . . + an
P
Exercise 3: Show that nk=0 nk = 2n . (Hint: As it’s written, this is an intimidating
formula to try and prove. So think about this problem first (which turns out to
be equivalent), you’re at a family reunion with n people and your family wants to
photograph every possible combination of people (including a photo with no one in
it). Show that they will have to take 2n photos.)
Note that a reformulation of the previous result is that the sum of the nth row of Pascal’s
triangle is 2n .
Exercise 4: Let Sn = 1 + 3 + 5 + . . . + (2n − 1) be the sum of the first n odd numbers.
Guess a closed formula for Sn and prove that it holds using induction.
n
Exercise 5: Show that n+1
= k−1
+ nk . (Hint: There is a symbol pushing way
k
of showing
this. However, the approach I hope you take goes back to the definition
n
of k . In fact, I challenge you to write a proof of this statement using only words
(no equations)).
Note that a reformulation of the previous result is the rule that the sum of two horizontally adjacent entries in Pascal’s triangle gives the entry underneath them.
Exercise
Binomial Theorem: Use induction to show that (x + y)n =
6 -k The
Pn
n
n−k
k=0 k x y
The binomial theorem will be very important in the coming weeks. For instance, we’ll
come back to it to find the derivative of f (x) = xn for n a positive integer.
n
k
1