Sequences
and Series  some ideas with Matlab

Sequences and series
are very related: a sequence of numbers
is a function defined on the set of positive integers (the numbers in
the sequence
are called terms).
In other words, a sequence is a list of numbers generated by
some mathematical rule and typically expressed in terms of n.
In order to construct the sequence, you group consecutive
integer values into n. 
For example,
the sequence of odd integers is generated by the sequence
2n – 1. The sequence produce 2(1)1,
2(2)1, 2(3)–1, 2(4)1... which
produces 1, 3, 5, 7...
A series is the sum of the terms
of a sequence. Series are
based on sums, whereas sequences are not.
In Matlab, let’s generate the sequence
above (for odd
numbers), in two ways:
a)
Using a forloop in a sequence:
%
We initialize our sum
s = 0;
%
We are going to get the first 5 terms,
% onebyone in our sequence
for n = 1 :
5
% This is the shown formula
for odd numbers
tn = 2*n1
% We addup every term in the
series
s = s + tn;
end
%
We display the sum of the series
s
The code
produces these results:
tn = 1
tn = 3
tn = 5
tn = 7
tn = 9
s = 25
b)
Using
a vectorized way, which is simpler and faster:
%
We define our first 5 ns
n
= 1 :
5
%
We calculate the terms
tn
= 2*n
 1
%
We addup all the terms in the series
s
=
sum(tn)
The code produces these
results:
n
= 1
2
3
4
5
tn = 1 3
5
7
9
s = 25
Arithmetic Sequences
This is a sequence of
numbers each of which (after the first)
is obtained by adding to the preceding number a constant called the
common
difference. Then, 3, 6, 9, 12, 15... is an arithmetic sequence because
each
term is obtained by adding 3 to the preceding number. In the arithmetic sequence
500, 450, 400... the common difference is 50.
Formulas for arithmetic sequences:
 The n^{th}
term, or last term: L = a + (n  1)d
 The sum of the first n
terms: S = n/2 (a + L) = n/2 [2a + (n  1)d]
where
a = first term of the sequence
d = common difference
n = number of terms
L = n^{th}
term, or last term
S = sum of first n
terms
Example:
Consider the arithmetic sequence 3, 7, 11, 15...
where a = 3 and d = 4.
The sixth term is:
L = a + (n  1)d = 3 +
(6  1)4 = 23.
The sum of the first six
terms is:
S = n/2(a + L) = 6/2(3 +
23) = 78 or
S = 6/2[2a + (n  1)d] = 6/2[2(3) + (6  1)4] = 78.
Let’s do it effortlessly
with Matlab, without forloops
(vectorized way):
%
Let's define our sequence, starting with 3,
%
using steps of four units, until we reach 999, for example
seq = 3
: 4 : 999;
%
Find the 6th element
sn =
seq(6)
%
Find the sum of the first 6 elements
s1_6
=
sum(seq(1 : 6))
The results are:
sn = 23
s1_6
=
78
Geometric Sequences
A geometric sequence is a
sequence of numbers each of which
is obtained by multiplying the preceding number by a constant number
called the
common ratio. So 4, 8, 16, 32... is a geometric sequence
because each number is
obtained by multiplying the preceding number by 2. In the geometric
sequence 64,
16, 4, l, 1/4... the common ratio is 4.
Formulas for geometric sequences:
 The n^{th}
term, or last term: L = ar^{n1}
 The sum of the first n terms:
S = a(r^{n}
– 1)/(r  1) = (rLa)/(r  1) (r and
1 are different)
where
a = first term
r = common ratio
n = number of terms
L = nth term, or last term
S = sum of first n terms
Example:
Consider the geometric
sequence 5, 10, 20, 40...
where a = 5 and r = 2
The seventh term is:
L = ar^{n1} = 5(2^{71})
= 5(2^{6}) = 320
The sum of the first
seven terms is:
S = a(r^{n1})/(r1) = 5(2^{7}1)/(21)
=
635
Let’s calculate it with Matlab:
%
Define your important constants
a = 5; r
= 2;
%
Define your sequence
n = 1 :
7;
L = a *
r.^(n1)
%
Find the 7th term
L(7)
%
Find sum of first 7 terms
sum(L(1:7))
Matlab response is:
L
=
5 10
20 40
80
160 320
ans
=
320
ans
=
635
Another way to approach
it is:
%
Define your exponents
n = 0 :
6
%
Define your sequence
gs =
2.^n * 5
%
Find the 7th term
gs(7)
%
Find the sum of the first 7 terms
sum(gs(1
: 7))
The answer is:
n =
0
1
2
3
4
5
6
gs =
5
10
20
40
80
160
320
ans = 320
ans = 635
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