Which choice gives an example that supports the conjecture, and a counterexample that shows the conjecture is false? For any real number n, vn² = n. a. V(9)? = 9 but V(2)² = 2 (-9)² = 9 but /(2)² = 2 (-9)² = 9 but /-2)² = 2 V(9)² = 9 but /(-2)² = 2 %3D %3D %3D %3D Od.

(12) Prove that if x EZ and r² is even, then x is even. (b) State and prove an analogous statement to (a) about odd numbers.

Find the interval in the real number line which equals U [4+20-1] n=1

X7. We know that given any real numbers x1 and x2, if x1X2 = 0, then x1 = 0 or x2 = 0. Since there are two factors, let's call this statement P(2). We assume P(2) is true without proving it. a. Use an inductive idea to carefully explain why given any real numbers x1, X2, and x3, if X1X2X3 = 0, then x1 = 0, x2 = 0, or X3 = 0. Be sure to use only P(2) in your argument. Hint: x,x2x3 = (x,X2)X3, which consists of two factors, x1X2, and x3. b. Use an inductive idea to carefully explain why given any real numbers x1, x2, X3, and x4, if X1X2X3X4 = 0, then x1 = 0, x2 = 0, x3 = 0, or x4 = 0. Be sure to use only P(2) or P(3) in %3D %3D your argument. c. Let P(n) be the statement that given any n real numbers X1, X2, ...,Xxn, if X1X2 Xn = 0, then x; = 0 for some i. Use mathematical induction to prove P(n) is true for all n 2 2. The base case does not need to be proven, but at least acknowledge its truth.

Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

(a) Recall that > 0 for every positive real number r. Prove that if a and b are positive real numbers, then 0 < Jab < b. (The number Jab is called the geometric mean of a and b, while (a + b)/2 is called the arithmetic mean or average of a and b.) (b) Under what conditions does Vab = (a +b)/2 for positive real numbers a and b? Justify your answer.

For integer values of a, use a proof by contraposition (also called contrapositive or indirect proof) to show: If a' +13 is even, then a is odd.

leoinov o and 7o iqeng od dw nislqont-( do (0) d o on tud pouso mbrore XE G Such that X? -3X -4=03 C{ imagin ary numbers } nombers True or FalseO why?

4. Prove that if n = nxn-1 holds for a positive integer value n, then it also holds for a negative integer value of -n, which means that x = -nx-n-1. Hint: quotient rule!

Is this an identity or a contradiction?