CS103A
HO# 44
Slides--Sequences, Induction
2/22/08
1
Logic and formal proofs
Proving mathematical theorems (number theory)
Crypto
Sequences and summations
Mathematical induction
Recursion
Combinatorics
Functions
CS103A
Sequences and Summations
A
sequence
is an ordered list, possibly infinite, of elements.
We will use the following notation:
a
1
,
a
2
,
a
3
, . . .
We also refer to the elements of the sequence as
terms
,
and if a
k
is a term, then k is its
index
or
subscript
.
An
explicit formula
for a sequence shows how to calculate
the value of each term from its subscript.
For example:
3,
6,
11,
18,
27, . . .
a
1
a
2
a
3
a
4
a
5
. . .
a
k
=
k
2
+ 2
Sequences and Summations
A
sequence
is an ordered list, possibly infinite, of elements.
We will use the following notation:
a
1
,
a
2
,
a
3
, . . .
We also refer to the elements of the sequence as
terms
,
and if a
k
is a term, then k is its
index
or
subscript
.
An
explicit formula
for a sequence shows how to calculate
the value of each term from its subscript.
For example:
3,
6,
11,
18,
27, . . .
a
1
a
2
a
3
a
4
a
5
. . .
a
k
=
k
2
+ 2
We can also specify a sequence by stating its starting value
and a
recursive formula
that tells us how to calculate a
k
from one or more preceding values.
a
k
= a
k-1
+ 2k – 1
where a
1
= 3
k:
1
2
3
4
5
6
7
8
a
k
:
7,
11,
15,
19,
23,
27,
31,
35 .
..
Explicit formula
a
k
=
Recursive formula
a
1
= 7
a
k
=
k:
1
2
3
4
5
6
7
8
9
10
a
k
:
0,
2,
8,
26,
80,
242,
728,
2186,
6560,
19682 .
..
Explicit formula
a
k
=
Recursive formula
a
1
= 0
a
k
=
Sum and Product Notation
Σ
k = 1
n
a
k
means
a
1
+ a
2
+ a
3
+ . . .
+ a
n
Π
k = 1
n
a
k
means
a
1
·a
2
3
· . . .
· a
n