Theoretical Framework
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ScribdTranslationsTheoretical Framework
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ScribdTranslationsWe can define a numeric sequence (or progression) as a set of
ordered numbers. Each of these numbers is called a term.
the succession:a1it is the first term, a2it is the second term, a3it's the third
term... anit is the n-th term.
Let's see the characteristics that define them:
Depending on the number they have, the sequences can
serfinitasoinfinitas.
The terms are increasing if each term is greater than its predecessor, that is,
an≤an+1
You have decremented us
an ≥ an+1
In the case of arithmetic and geometric sequences, we can find a
formula, which we call the general formula of the progression, which indicates
the value of any term in the sequence without the need to write the
previous terms. Likewise, we can calculate the sum of n terms.
consecutive and, at times, the sum of infinite terms.
GENERAL TERM
The general term of a sequence is the formula anwhich allows knowing each
term in terms of its position n.
Examples:
The general term of the progression of odd numbers (1, 3, 5,
7,... is
an= 2⋅n - 1
We use the general term to calculate some of its terms.
substituting the position n:
a1= 2⋅1−1 = 1
a2=2⋅2−1 = 3
a3= 2⋅3−1 = 5
The general term of the sequence 1, 4, 9, 16, 25, 36,... is
an= n2
ARITHMETIC PROGRESSION
A progression is arithmetic when each term is the sum of the term
anterior plus a constant number, which we call difference or ratio
Examples:
100 105 110 115 120 …
It is an arithmetic sequence whose difference or ratio is d = 5
-5 -3 -1 1 3 5
It is an arithmetic sequence whose difference or ratio is 2
1 4 9 16 25 36
It is not an arithmetic secession, because, although the second
the term is obtained by adding 3, the same does not happen with the
following terms.
The general formula of the arithmetic progression is:
anan + d
GEOMETRIC PROGRESSION
A progression is geometric if each term is obtained by multiplying a
constant number (ratio) by the previous term.
Examples:
1, 3 9 27 81, ...
It is a geometric progression whose ratio is r = 3.
6, 12 24, 48 96,…
It is a geometric sequence whose ratio is r = 2.
5 25, 50 150,…
It is not a geometric sequence because, although the second term is obtained
multiplying the first by 5, the same does not happen with the following.
The general formula of the geometric progression is:
aa⋅r
The complementary formulas are:
ARITHMETIC SEQUENCE
It is in the form
Difference General Term
Sum of the first terms
GEOMETRIC SEQUENCE
It is in the form
a1
a2=a1. r
a3= a2. r
a4= a3. r
Reason General term
Sum of the first terms Sum of all the terms
a 1= an /rn-1