1. Use mathematical induction to show that, ∀n ≥ 3,
2n2 + 1 ≥ 5n
2....
1. Use mathematical induction to show that, ∀n ≥ 3,
2n2 + 1 ≥ 5n
2. Letting s1 = 0, find a recursive formula for the
sequence 0, 1, 3, 7, 15,...
3. Evaluate. (a) 55mod 7. (b) −101 div 3.
4. Prove that the sum of two consecutive odd integers is
divisible by 4
5. Show that if a|b then −a|b.
6. Prove or disprove: For any integers a,b, c, if a ∤ b and b ∤
c, then...
Consider sequences of n numbers, each in the set {1, 2, . . . ,
6}...
Consider sequences of n numbers, each in the set {1, 2, . . . ,
6}
(a) How many sequences are there if each number in the sequence
is distinct?
(b) How many sequences are there if no two consecutive numbers
are equal
(c) How many sequences are there if 1 appears exactly i times in
the sequence?
1. Consider the sequences defined as follows.(an)
=(12,13,23,14,24,34,15,25,35,45,16,26,36,46,56,17, . . .),(bn)
=(n2(−1)n)= (−1,4,−9,16, . . .).(i)...
1. Consider the sequences defined as follows.(an)
=(12,13,23,14,24,34,15,25,35,45,16,26,36,46,56,17, . . .),(bn)
=(n2(−1)n)= (−1,4,−9,16, . . .).(i) For each sequence, give its lim
sup and its lim inf. Show your reasoning; definitions are not
required.(ii) For each sequence, determine its set of subsequential
limits. Proofs are not required.
1. Write the explicit rules for the following sequences:
A) 1/2, 2/5, 3/10, 4/17, 5/26, 6/37,........
1. Write the explicit rules for the following sequences:
A) 1/2, 2/5, 3/10, 4/17, 5/26, 6/37,.....
B) 3, -5/1, 7/6, 9/24, 11/120,...
2. Given two terms of the arithmetic sequence find the common
difference, the first term, and the explicit formula ( a subscript
n)
a) a subscript 17= 105
a subscript 40 = -30
b) a subscript 10
a subscript 40= 100
b)