Document Preview: 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 XXXXXXXXXXpn= knk=1 (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 j=1n j3 = XXXXXXXXXXn3 =
[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 2n <>