Successions

A succession is an ordered sequence of numbers, such as:

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A succession of real numbers is an application of the set N (set of natural number excluding zero) in the set R of the real numbers.  

  It is called term of a succession to each one of the elements that make un the succession. To represent the different terms of a sequence, the same letter is used with different subscripts, which indicate the place which that therm occupies in the sequence.  

For example: 

  • In the succession: a) 1, 2, 3, 4, 5, 6,… we have that: a5 = 5, because it is the term of the succession that occupies the fifth place.  
  • In the succession: b) 2, 4, 6, 8 , 10,… the third term would be written by b3 and it would corresponded to the value 6.  

   The really important thing when it comes to naming the terms of a sequence is the subscript because it denotes the place it occupies in the sequence. The letters with which the succession is designated are different for different sequences and are usually lowercase letters. 

   We called general term of a succession to the term that occupies the n-th and it is written with the letter that denotes the succession (for example a) with subscript n:(an).

If we focus on the values ​​taken by the subscripts, we see that they are natural numbers, but the terms of the sequence do not have to be so, that is, the values ​​taken by the sequence are real numbers. Therefore, we can affirm that a succession of real numbers is an application that corresponds to each natural number a real number.  

Ways to define a succession

There are several ways to define a succession:

  • Giving its general term: A sequence has infinito terms and is frequently expressed by its general term an, that since an is a function that depends on n, it is enough to give natural values to the indeterminate n to obtain any term of the sequence.
    For example: the sequence has as a general rule:  
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We can form the terms of the sequence by successively giving “n” los values 1, 2, 3…, having then:  

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  • Giving a property that meets the terms of the succession: If we insure a property that meets the succession, it is easier to know the value of an. 
    – Ex: thanks to the following properties, get the successions:  

    1. Succession of pair numbers: 2, 4, 6, 8, …
    2. Succession of prime numbers: 2, 3, 5, 7, 11, …
    3. Succession of the natural numbers ending in 9: 9, 19, 29, 39, …
  • By a law of recurrence: you can get a term from previous ones.  – Ex: Knowing that the first term of a succession is 2, and each term, except the first one, is three times the previous term, write the first terms of the sequence.  
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    It is said that a succession of real numbers is increasing when it is verified that each term is less than or equal to the next one, that is:  

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    On the other side, it is said that a succession is decreasing if each term is greater than or equal to the next one: 

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Arithmetic progressions

An arithmetic progression is a succession of real numbers in which the difference between two consecutive terms in the sequence is constant. This constant is called “difference of progression” and is usually denoted by the letter d.  

   In an arithmetic progression, where i is any natural number, it is verified that:

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That is, each term is obtained by adding the difference to the previous one, d:

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Depending on the value of the difference “d”, we can find different types of arithmetic sequences.  

  • If d>0, the succession is increasing, that is, each term is greater than the previous ones.  
  • If d<0, the succession is decreasing, each term being smaller than the previous ones.  
  • If d=0, the sequence is constant and all its terms are equal. 

For example: 

If a1=3 and d=2, What will be the first five terms of the arithmetic progression?

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General term of an arithmetic progression

As happens with successions, a progression is perfectly defined if we know its general term. But, if we wanted to find the term 50, its calculation would be very uncomfortable finding all the numbers, therefore, we are interested in finding the expression of the general term.  

   Knowing the values ​​of a1 and d, and taking into account the definition of arithmetic progression, we can obtain the general term of the same. 

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Generally:

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Therefore, the general term of an arithmetic progression is:

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Generalizing this result, we can calculate the general term of an arithmetic progression knowing d and another term of the progression, not necessarily the first one. 

   More general, being ak the term of the progression that occupies place k, the general term of an arithmetic progression is: 

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Depending on the data we have, we can calculate the general term of an arithmetic progression in one way or another, because: 

  • If we know a1 and d, we can apply:
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  • If we know any term ai and d, we can use:
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  • If we know two any terms ar and as, by means of the equations:  
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       we can clear d as a function of r, s, ar and as and therefore, we have:

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For example:  

Knowing that a10=24 y d=3; find the value of a40.

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Interpolation of n arithmetic means

We call interpolating “h arithmetic means” between two numbers given “p and q” to intercalate h terms between p and q so that they are in arithmetic progression, with p and q being the extremes.

p …………….. (h terms) ………………… q

For example:  

In the case of the arithmetic progression: 2, a, b, c, 14

We can see that a1 = 2 and a5 =14 so the value of p is the first value and q will be the term h+2.

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  The difference of the progression is clearing d from the general expression.

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Where: 

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Knowing the difference, the arithmetic means can be easily obtained, which will be:

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Sum of the consecutive terms of a limited arithmetic progression

Due to the arithmetic progressions have infinite terms, we can do the sum of the consecutive terms of a limited arithmetic progression in the following way:  

Being the limited arithmetic progression:

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The sum of these n terms will be:

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And too:

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Adding member to member the two previous equalities results:  

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Grouping and finally clearing, we have: 2Sn = (a1 + an)n

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Therefore, the sum of the terms of a limited arithmetic progression equals the half-sum of the extremes by the number of terms.  

Geometric progressions

A geometric progression is a succession of real numbers in which the quotient between each term and the previous one is constant. This constant is called the ratio of progression and it is usually denoted by the r. That is, if i is a natural number and whenever ai is non-zero, the value of r is: 

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Or what is the same, each term is obtained by multiplying the previous one for the reason r:

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General term of a geometric progression

A geometric progression, due to it is a succession, is totally defined if we know its general term. We are going to obtain it without more than to apply the definition of geometric progression: 

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Therefore, the general term of a geometric progression is:

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Generalizing this result, we can calculate the general term of a geometric progression knowing r and another term of the progression, not necessarily the first one. Being ak the term of the progression that occupies place k, the general term of a geometric progression is:

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Depending on the value of r, we can find different types of geometric progressions:  

  • If r >1, the progression is growing, that is, each term is greater than the previous ones. {2, 4, 8, 16…}
  • If 0< r <1, the progression is decreasing, that is, each term is smaller than the previous ones. {90, 30, 10,…}
  • If r <0, the progression is alternate, that is, its terms change their sign according to the value of n. {-2, 4, -8, 16…}
  • If r = 0, the progression is the progression formed by zeros from the second term. {7, 0, 0, 0,…}
  • If r = 1, the progression is the constant progression formed by the first term.{2, 2, 2, 2,…}

Depending on the data we have, we can calculate the general term of a geometric progression in one way or another: 

  • If we know a1 and r, we can apply: an = a1 · r n-1
  • If we know any term ak and r, we can use: an = ak · r n-k. 
  • If we know two any tems ap and aq, with ap no null, we need to know the reason r to be able to apply the previous reason. But we know that: an = ap · r n-p   and   an = aq · r n-q . We can clear r according to p, q, ap and aq and we have finally:  
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Product of the terms of a geometric progression

In a geometric progression, the producto of two equidistante terms is constant. That is, if the natural subscripts p, q, t and s verify that p+q = t+s, then: ap · aq = at · as because: 

ap·aq = a1 · rp-1·a1 · rq-1 = a12 ·rp-1 ·rq-1 = a12 · rp+q-2

at·as = a1 · rt-1·a1 · rs-1 = a12 ·rt-1 ·rs-1 = a12 · rt+s-2

And as: p+q = t+s, so: ap · aq= at · as

We can calculate the product of the n terms of a geometric progression, Pn. That is: 

Pn= a1 · a2 · a3 ·……….· an-2 · an-1 · an

Applying the conmmtative property of the product, we have:  

     Pn = an · an-1 · an-2 · ……..· a3 · a2· a1

Multiplying these two equalities, we have:  

     Pn2 = (a1 · a2· a3·………· an-2· an-1·an) · (an· an-1· an-2·……· a3 · a2· a1)

     Pn2 = (a1 · an) · (a2· an-1) · (a3· an-2)·……· (an-2 · a3) · (an1 · a2) · (an·a1)

As we can see, the subscripts corresponding to each pair of terms in parentheses add n + 1, so the product will always be the same in each factor, then: Pn2 = (a1 · an)n

The product of the first n terms of a geometric progression is given by: 

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Sum of the terms of a geometric progression

The sum of the first n terms of a geometric progression, provided that r is different from 1, is given by: 

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According to the values of r, the value of Sn will be: 

  • If |r| > 1, the terms in absolute value grow indefinitely and the value of Sn s given by the previous formula. 
  • If |r| < 1, the sum of its terms when n is large, Sn approaches to a a/(1-r), since if in
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    we raise the reason |r|<1 at a power, the greater is the exponent, the smaller is the value of rn and if n is large enough, rn approaches 0. Therefore, 

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  • If r = – 1, the consecutive terms are opposite: {a1, -a1, a1, -a1, ….} and Sn is equal to zero if n is even, and equal to a1 if n is odd. The sum of the series oscillates between these two values.
     

But could we add an unlimited number of consecutive terms of a geometric progression? Depending on the value of r it will be possible or not to obtain the sum of an unlimited number of terms: 

  • If r = 1, the progression is the constant progression formed by the first term: {a1, a1, a1, a1, …} and if a1 s positive, the sum of the terms will be increasing (if it were a1 negative it would be the increasing sum in absolute value, but negative). Therefore, if the number of terms is unlimited, this sum will be infinite. 
  • If |r| > 1, the terms grow indefinitely and the value of Sn for an unlimited number of terms, will also be infinite. 
  • If |r| < 1, the sum of its terms is approximated when n is large to a1/(1-r)
  • If r = -1, the consecutive terms are opposite: {a1, -a1, a1, -a1, … } and Sn is equal to zero if n is even, and equal to a1 if n is odd. The sum of the series oscillates between these two values for a finite number of terms. For an unlimited number of terms we don not know if it is even or odd, so that the sum can not be realized unless that a1=0, case in which S = 0 = a1/(1-r). In the rest of the cases we say that the sum of infinite terms does not exist because its value is oscillating. 
  • If r < -1, the terms oscillate between positive and negative values, growing in absolute value. The sum of its infinite terms does not exist because its value is also oscillating. 

Therefore, we can affirm that the sum of an unlimited number of terms of a geometric progression only takes a finite value if |r| < 1, and then it is given by:

 

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In the rest of the cases, or it is infinite, or it does not exist as it oscillates. 

For example: 

Calculate the sum of all the terms of the geometric progression whose first term is 4 and the ratio is 1/2.

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Applications of the geometric progressions

Generating fraction

An exacto or periodic number can be expressed as a fraction, “generating fraction” as follows:

  • Calculation of the generation fraction of a pure periódico decimal expression.
    We try to find a fracción such that dividing the numerator between the denominator gives us the starting number. That is: 

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    We observe that it is the sum of the infinite terms of an unlimited geometric progression of ratio 1/100. Thus: 

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Calculation of the generativa fracción of a mixed periódico decimal.
In this case, the easiest is to decomise the expression as the sum of the two decimal expressions. That is:

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Compound capitalization

If we depositó in a Financial entity a quantity of money C0 during a time t and a credit r given in as much by one, we will obtain a benefit known as interes, and given by the formula:

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  The main characteristic of the compound capitalization is that the interest generated in a year, become part of the initial capital and produce interest in the following periods.  

   The final capital obtained after n years given an initial capital C0 and a credit r given in as much by one, is: 

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