Como Calcular El Termino General De Una Progresion Geometrica último 2023

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

$displaystyle a_n$

to the term that occupies the position

$displaystyle n$

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

$displaystyle a_1$

) and reason (

$displaystyler$

) using the following formula called General term:

$displaystyle a_n=a_1cdot r^n-1,$

Examples of geometric progressions

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

  

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

  .

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

  . 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 (

$displaystyle b_n$

) defined by the conditions

${displaystyle b_n=leftbeginarrayllclsi&n=1&longrightarrow &psi&n>1&longrightarrow &b_n-1cdot qendarray right.$

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

called recursive equation of order 1 ​(

$displaystyle qneq 0$

),

$displaystyle n=1,2,…$

(

$displaystyle q$

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 (

$displaystyle a_ngeq a_n-1$

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

$displaystyle a_nleq a_n-1$

), constant when all terms are equal (

$displaystyle a_n=a_n-1$

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

$displaystyle r<0$

).​

Monotonicity as a function of the first term,

$displaystyle a_1$

and of reason,

$displaystyler$

:​

$displaystyle a_1>0$ 0″>	$displaystyle r>1$ 1″>	growing
	$displaystyle 0$	decreasing
$displaystyle a_1<0$	$displaystyle r>1$ 1″>	decreasing
	$displaystyle 0$	growing
	$displaystyler=1$	constant
	$displaystyle r<0$	alternated

Sum of terms of a geometric progression

Sum of the first n terms of a geometric progression

It is denoted by

$displaystyle S_n$

to the sum of the

$displaystyle n$

first consecutive terms of a geometric progression:

$displaystyle S_n=a_1+a_2+…+a_n-1+a_n$

This sum can be calculated from the first term

$displaystyle a_1$

and of reason

$displaystyler$

using the formula

$displaystyle S_n=a_1cdot frac r^n-1r-1$

Be $displaystyle S_n=a_1+a_2+…+a_n-1+a_n$ Both members of the equality are multiplied by the ratio of the progression $displaystyler$ . $displaystyle rcdot S_n=rcdot (a_1+a_2+…+a_n-1+a_n)$ $displaystyle rcdot S_n=rcdot a_1+rcdot a_2+…+rcdot a_n-1+rcdot a_n$ since $displaystyle rcdot a_i=a_i+1$ $displaystyle rcdot S_n=a_2+a_3+…+a_n+a_n+1$ If we proceed to subtract from this equality the first one: $displaystyle rcdot S_n-S_n=a_n+1-a_1$ since all intermediate terms mampara each other out. clearing $displaystyle S_n$ : $displaystyle S_n=frac a_n+1-a_1r-1=frac a_1cdot r^n-a_1 r-1=a_1cdot frac r^n-1r-1$ In this way, the sum of the $displaystyle n$ 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 $displaystyle a_n$ What: $displaystyle S_n=frac a_1cdot r^n-a_1r-1=a_1cdot frac r^n -1r-1$ which expresses the sum of $displaystyle n$ consecutive terms of a geometric progression as a function of the first term and the ratio of the progression.

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

$displaystyle a_m$

Y

$displaystyle a_n$

(both included):

$displaystyle sum _k=m^na_k=frac rcdot a_n-a_mr-1=a_1cdot frac (r^n-r^m-1)r-1=a_mcdot frac (r^n-m+1-1) r-1$

Sum of infinite terms of a geometric progression

If the absolute value of the ratio is less than unity

$<1$

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

$<1$

,

$displaystyle r^infty$

tends towards 0, so that:

$displaystyle S_infty =a_1cfrac r^infty -1r-1=a_1cdot cfrac 0-1r- 1$

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

$displaystyle S_infty =cfrac a_11-r$

,

$<1$

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

$displaystyle n$

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

$displaystyle prod _i=1^na_i=left(sqrt a_1cdot a_nright)^n$

(Yeah

$displaystyle a_1,r>0$

0″>).

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

$displaystyler$

(Yeah

$displaystyle a_1,r>0$

0″>), are in arithmetic progression of difference

$displaystyle log r$

one has:

$displaystyle log(prod _i=1^na_i)= sum _i=1^nlog a_i= frac ( log a_1+log a_n)n2= log left(sqrt a_1cdot a_nright)^n$

,

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

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