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During the study of discrete mathematics, I found this course very informative and applicable.The main points in these lecture slides are:Sequences and Summations, Sequence Examples, Geometric Vs Arithmetic Sequences, Constant Factor, Fibonacci Sequence, Fibonacci References, Golden Ratio, Determining Sequence Formula, Geometric Progression
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1
An important task of mathematics is to discover and
characterize regular patterns, such as those
associated with repeated processes.
processes is the sequence.
about patterns in sequences is mathematical
induction.
2
4
shows how the values of a (^) k depend on k.
5
Let m and n be integers such that m ≤ n.
Then
We call k index of the summation; m the lower limit of the summation; n the upper limit of the summation.
Ex.: Suppose a 3 =2, a 4 =-4, a 5 =0, a 6 =7. Then
m m n
n
k m
3 4 5 6 2 (^4 )^075
6
3
=
a a a a a k
k
7
Let m and n be integers such that m ≤ n.
Then
Examples : ♦
♦ For each n∈ Z +^ , is called n factorial. E.g., 4! = 1 · 2 · 3 · 4 = 24 Note: 0! = 1
m m n
n
k m
4
1 4
3 3
2 2
1 1
3
1
= ⋅ ⋅ =
k = k
k
1 2! 1
k n n
n
k
=
8
Recall that
if a = pk · 2k^ + pk-1 · 2k-1^ + … + p 1 · 2^1 + p 0 · 2^0 then a 10 = (pk pk-1 … p 1 p 0 ) 2 , where p 0 , p 1 , …, pk-1 , pk is a sequence of binary digits 0 and 1.
Question: How to find p 0 , p 1 , …, pk-1 , pk?
10
Thus, 14 10 = 1110 2
Generally , to get binary representation
for nonnegative integer a,
