Como Calcular El Termino General De Una Progresion Geometrica último 2023

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Como Calcular El Termino General De Una Progresion Geometrica

A geometric progression is a sequence of experimental numbers called terms, in which each term is obtained by multiplying the previous term by a constant called reason either hábitat of progression. If it is denoted by

anodisplaystyle a_n

to the term that occupies the position

nodisplaystyle n

of the sequence, the value of any term can be obtained from the first (

a1displaystyle a_1

) and reason (

rdisplaystyler

) using the following formula called General term:

ano=a1rno1displaystyle a_n=a_1cdot r^n-1,

Examples of geometric progressions

  • The progression 5, 15, 45, 135, 405,…’ is a geometric progression for good reason.
    r=3displaystyle r=3

  • The progressions 1, 2, 4, 8, 16,… and 5, 10, 20, 40,… are geometric with reason
    r=2displaystyle r=2

    .

  • The progression -3, 6, -12, 24, … is right
    /r=2displaystyle /r=-2

    . This progression is also an alternate succession.

  • Other examples are: the paradox of Achilles and the tortoise, the problem of wheat and the chessboard, and the number of movements of the rings in the tower of Hanoi.​

recursive definition

Is named geometric progression a number sequence (

bnodisplaystyle b_n

) defined by the conditions

bno={yesYono=1pyesYono>1bno1what{displaystyle b_n=leftbeginarrayllclsi&n=1&longrightarrow &psi&n>1&longrightarrow &b_n-1cdot qendarray right.

1&longrightarrow &b_n-1cdot qendarray right.»>

called recursive equation of order 1 ​(

whatdisplaystyle qneq 0

),

no=1,2,...displaystyle n=1,2,…

(

whatdisplaystyle q

is the ratio of geometric progression)

Monotony

A geometric progression is monotone increasing when each term is greater than or equal to the previous one (

anoano1displaystyle a_ngeq a_n-1

), monotone decreasing when each term is less than or equal to the previous one (

anoano1displaystyle a_nleq a_n-1

), constant when all terms are equal (

ano=ano1displaystyle a_n=a_n-1

) Y alternated when each term has a different sign than the previous one (occurs when

r<displaystyle r<0

).​

Monotonicity as a function of the first term,

a1displaystyle a_1

and of reason,

rdisplaystyler

:​

a1>displaystyle a_1>0

0″>

r>1displaystyle r>1

1″>

growing

<r<1displaystyle 0

decreasing

a1<displaystyle a_1<0

r>1displaystyle r>1

1″>

decreasing

<r<1displaystyle 0

growing

r=1displaystyler=1

constant

r<displaystyle r<0

alternated

Sum of terms of a geometric progression

Sum of the first n terms of a geometric progression

It is denoted by

Snodisplaystyle S_n

to the sum of the

nodisplaystyle n

first consecutive terms of a geometric progression:

Sno=a1+a2+...+ano1+anodisplaystyle S_n=a_1+a_2+…+a_n-1+a_n

This sum can be calculated from the first term

a1displaystyle a_1

and of reason

rdisplaystyler

using the formula

Sno=a1rno1r1displaystyle S_n=a_1cdot frac r^n-1r-1

Be

Sno=a1+a2+...+ano1+anodisplaystyle S_n=a_1+a_2+…+a_n-1+a_n

Both members of the equality are multiplied by the ratio of the progression

rdisplaystyler

.

rSno=r(a1+a2+...+ano1+ano)displaystyle rcdot S_n=rcdot (a_1+a_2+…+a_n-1+a_n)

rSno=ra1+ra2+...+rano1+ranodisplaystyle rcdot S_n=rcdot a_1+rcdot a_2+…+rcdot a_n-1+rcdot a_n

since

raYo=aYo+1displaystyle rcdot a_i=a_i+1

rSno=a2+a3+...+ano+ano+1displaystyle rcdot S_n=a_2+a_3+…+a_n+a_n+1

If we proceed to subtract from this equality the first one:

rSnoSno=ano+1a1displaystyle rcdot S_n-S_n=a_n+1-a_1

since all intermediate terms mampara each other out.

clearing

Snodisplaystyle S_n

:

Sno=ano+1a1r1=a1rnoa1r1=a1rno1r1displaystyle S_n=frac a_n+1-a_1r-1=frac a_1cdot r^n-a_1 r-1=a_1cdot frac r^n-1r-1

In this way, the sum of the

nodisplaystyle n

terms of a geometric progression when the first and last terms of the same are known. If you want to simplify the formula, you can express the universal term of the progression

anodisplaystyle a_n

What:

Sno=a1rnoa1r1=a1rno1r1displaystyle S_n=frac a_1cdot r^n-a_1r-1=a_1cdot frac r^n -1r-1

which expresses the sum of

nodisplaystyle n

consecutive terms of a geometric progression as a function of the first term and the ratio of the progression.

350px Geometric progression convergence diagram.svg

Geometric series 1 + 1/2 + 1/4 + 1/8 + … converges to 2.

The previous procedure can be generalized to obtain the sum of the consecutive terms included between two arbitrary elements

amdisplaystyle a_m

Y

anodisplaystyle a_n

(both included):

what=mnoawhat=ranoamr1=a1(rnorm1)r1=am(rnom+11)r1displaystyle sum _k=m^na_k=frac rcdot a_n-a_mr-1=a_1cdot frac (r^n-r^m-1)r-1=a_mcdot frac (r^n-m+1-1) r-1

Sum of infinite terms of a geometric progression

If the absolute value of the ratio is less than unity

|r|<1<1

, the sum of the infinitely many decreasing terms of the geometric progression converges towards a finite value. Indeed, yes

|r|<1<1

,

rdisplaystyle r^infty

tends towards 0, so that:

S=a1r1r1=a11r1displaystyle S_infty =a_1cfrac r^infty -1r-1=a_1cdot cfrac 0-1r- 1

Finally, the sum of the infinite terms of a geometric progression with a ratio less than unity is:

S=a11rdisplaystyle S_infty =cfrac a_11-r

,

|r|<1<1

presticioso case

An example of geometric progression appears in the case of one of Zeno’s paradoxes: the challenge of Achilles and the tortoise.

product of the first no terms of a geometric progression

The product of the

nodisplaystyle n

first terms of a geometric progression can be obtained by the formula

Yo=1noaYo=(a1ano)nodisplaystyle prod _i=1^na_i=left(sqrt a_1cdot a_nright)^n

(Yeah

a1,r>displaystyle a_1,r>0

0″>).

Since the logarithms of the terms of a geometric progression of ratio

rdisplaystyler

(Yeah

a1,r>displaystyle a_1,r>0

0″>), are in arithmetic progression of difference

logrdisplaystyle log r

one has:

log(Yo=1noaYo)= Yo=1nologaYo= (loga1+logano)no2= log(a1ano)nodisplaystyle log(prod _i=1^na_i)= sum _i=1^nlog a_i= frac ( log a_1+log a_n)n2= log left(sqrt a_1cdot a_nright)^n

,

and taking antilogarithms the formula is obtained.

See also

  • arithmetic progression
  • geometric series

References

external links

  • Weisstein, Eric W. «Geometric Progression.» In Weisstein, Eric W., ed. MathWorld (in English). Wolfram Research.
  • Sum of Geometric Progression Calculator
authority gimnasia
  • Wikimedia projects
  • wd Data: Q173523
  • Dictionaries and encyclopedias
  • Britannica: url
  • wd Data: Q173523


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