Exercise #12) Show that [x]+ [x + 1/2]= [2x] whenever x is a real number. Exercise #22) Conjecture a formula for the nth term of {an} if the first ten terms of this sequence are as follows: a)

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If necessary, they need to be done using Maple software or Microsoft Equation 3.0

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Exercise #12) Show that [x]+ [x + 1/2]= [2x] whenever x is a real number. Exercise #22) Conjecture a formula for the nth term of {an} if the first ten terms of this sequence are as follows: a) 3, 11, 19, 27, 35, 43, 51, 59, 67, 75 b) 5, 7, 11, 19, 35, 67, 131, 259, 515, 1027 c) 1, 0, 0, 1, 0, 0, 0, 0, 1, 0 d) 1, 3, 4, 7, 11, 18, 29, 47, 76, 123 Exercise #6) By putting together two triangular arrays, one with n rows and one with n – 1 rows, to form a square (as illustrated for n = 4), show that tn-1 + tn = n2, where tn is the nth triangular number. (it’s a 4 by 4 square with 16 dots in it)- fyi Exercise #10) Show that p1=1and pk=pk-1+(3k -2) for k =2. Conclude that pn= (3k -2) and evaluate this sum to find a simple formula for pn. Exercise #18) Find n! for n equal to each of the first ten positive integers. Exercise #8) Use mathematical induction to prove that = + + . . . + = [n(n + 1)/2]² for every positive integer n. Exercise # 14) Show that any amount of postage that is an integer number of cents greater than 53 cents can be formed using just 7-cent and 10-cent stamps. Exercise # 20) Use mathematical induction to prove that < n! for n = 4. Exercise # 34) Use mathematical induction to show that a × chessboard with one square missing can be covered with L-shaped pieces, where each L-shaped piece covers three squares. Exercise #2) Find each of the following Fibonacci numbers. a) f12 b) f16 c) f24 d) f30 e) f32 f) f36 Exercise #6) Prove that fn-2 + fn+2 = 3fn whenever n is an integer with n = 2. (Recall that f0 = 0.) Exercise #10) Prove that f2n+1= n+1+ n whenever n is a positive integer. Exercise #40) Show that if an= (an – ßn), where a = (1+v5)/2 and ß = (1-v5)/2, then an=an-1+ an-2 and a1= a2 = 1. Conclude that fn= an, where fn is the nth Fibonacci number. A linear homogeneous recurrence relation of degree 2 with constant coefficients is an equation of the form an= c1an-1+ c2an-2, where c1 and c2 are real numbers w…

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