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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