This document discusses using Geogebra to model recursive sequences, recursive successions, and recursive progressions. It provides examples of one-parameter and two-parameter recursive functions defined using Geogebra's sequence, iteration list, and zip functions. The examples include linear, exponential, trigonometric, fractional, and other recursive functions over varying number of terms.

1. Secuencias Recursivas, Sucesiones Recursivas & Progresiones Recursivas
con Geogebra
Inglés = Español Inglés = Español
Execute = Ejecuta Sequence = Secuencia
Join = Encadena IterationList = ListaIteración
Un parámetro de recursión con la variable de iteración i
Execute[Join[{"f_{1} =-1"}, Sequence["f_{"+(i)+"}="+i+"*f_{"+(i-1)+"}",i, 2, 10]]]
Ejecuta[Encadena[{"f_{1} =-1"}, Secuencia["f_{"+(i)+"}="+i+"*f_{"+(i-1)+"}",i, 2, 10]]]
Execute(Join({{"l2={-1}"}, Sequence("SetValue(l2," + (n) + "," + (n) + "* l2(" + (n) + "-1))", n, 2, 10)}))
Ejecuta(Encadena({{"l2={-1}"}, Secuencia("Valor(l2," + (n) + "," + (n) + "* l2(" + (n) + "-1))", n, 2, 10)}))
Zip(Element(L, 2), L, IterationList({Element(lst, 1) + 1, (Element(lst, 1) + 1) * Element(lst, 2)}, lst, {{1, -1}}, 9))
Zip(Elemento(L, 2), L, ListaIteración({Elemento(lst, 1) + 1, (Elemento(lst, 1) + 1) * Elemento(lst, 2)}, lst, {{1, -1}}, 9))
Execute[Join[{"f_{1} =2"}, Sequence["f_{"+(i)+"}=f_{"+(i-1)+"}+"+i,i, 2, 10]]]
Ejecuta[Encadena[{"f_{1}=2"},Secuencia["f_{"+(i)+"}=f_{"+(i-1)+"}+"+i,i, 2, 10]]]
Execute(Join({{"l2={2}"}, Sequence("SetValue(l2," + (n) + "," + (n) + " + l2(" + (n) + "-1))", n, 2, 10)}))
Ejecuta(Encadena({{"l2={2}"}, Secuencia("Valor(l2," + (n) + "," + (n) + " + l2(" + (n) + "-1))", n, 2, 10)}))
Zip(Element(L, 2), L, IterationList({Element(lst, 1) + 1, Element(lst, 1) + 1 + Element(lst, 2)}, lst, {{1, 2}}, 9))
Zip(Element(L, 2), L, ListaIteración({Elemento(lst, 1) + 1, Elemento(lst, 1) + 1 + Elemento(lst, 2)}, lst, {{1, 2}}, 9))
Execute[Join[{"f_{1}=2"},Sequence["f_{"+(i)+"}=- f_{"+(i-1)+"}*(1+1/"+i+")",i, 2, 10]]]
Ejecuta[Encadena[{"f_{1}=2"},Secuencia["f_{"+(i)+"}=- f_{"+(i-1)+"}*(1+1/"+i+")",i, 2, 10]]]
Execute(Join({{"l2={2}"}, Sequence("SetValue(l2," + (n) + ",-(1+1/" + (n) + ") * l2(" + (n) + "-1))", n, 2, 10)}))
Ejecuta(Encadena({{"l2={2}"}, Secuencia("Valor(l2," + (n) + ",-(1+1/" + (n) + ") * l2(" + (n) + "-1))", n, 2, 10)}))
Zip(Element(L, 2), L, IterationList({Element(lst, 1) + 1, -(1+1/(Element(lst, 1) + 1)) * Element(lst, 2)}, lst, {{1, 2}}, 9))
Zip(Elemento(L, 2), L, ListaIteración({Elemento(lst, 1) + 1, -(1+1/(Elemento(lst, 1) + 1)) * Elemento(lst, 2)}, lst, {{1, 2}}, 9))
Secuencias Recursivas con Geogebra Prof. Pérez Sanabria

2. Execute[Join[{"f_{1}=2"},Sequence["f_{"+(i)+"}= f_{"+(i-1)+"}*(5*"+i+"+1)/(4*"+i+"+3)",i, 2, 10]]]
Ejecuta[Encadena[{"f_{1}=2"},Secuencia["f_{"+(i)+"}= f_{"+(i-1)+"}*(5*"+i+"+1)/(4*"+i+"+3)",i, 2, 10]]]
Execute(Join({{"l2={2}"}, Sequence("SetValue(l2," + (n) + ",(5*"+(n)+"+1)/(4*" + (n) + "+3) * l2(" + (n) + "-1))", n, 2, 10)}))
Ejecuta(Encadena({{"l2={2}"}, Secuencia("Valor(l2," + (n) + ",(5*"+(n)+"+1)/(4*" + (n) + "+3) * l2(" + (n) + "-1))", n, 2, 10)}))
Zip(Element(L, 2), L, IterationList({Element(lst, 1) + 1, ((5*(Element(lst, 1) + 1)+1)/(4*(Element(lst, 1) + 1)+3)) * Element(lst, 2)}, lst, {{1,
2}}, 9))
Zip(Elemento(L, 2), L, ListaIteración({Elemento(lst, 1) + 1, ((5*(Elemento(lst, 1) + 1)+1)/(4*(Elemento(lst, 1) + 1)+3)) * Elemento(lst, 2)},
lst, {{1, 2}}, 9))
Execute[Join[{"f_{1}=1"},Sequence["f_{"+(i)+"}= f_{"+(i-1)+"}*(2+cos("+i+"))/sqrt("+i+")",i, 2, 10]]]
Ejecuta[Encadena[{"f_{1}=1"},Secuencia["f_{"+(i)+"}= f_{"+(i-1)+"}*(2+cos("+i+"))/sqrt("+i+")",i, 2, 10]]]
Execute(Join({{"l2={1}"}, Sequence("SetValue(l2," + (n) + ",(2+cos("+(n)+"))/(sqrt(" + (n) + "))* l2(" + (n) + "-1))", n, 2, 10)}))
Ejecuta(Encadena({{"l2={1}"}, Secuencia("Valor(l2," + (n) + ",(2+cos("+(n)+"))/(sqrt(" + (n) + "))* l2(" + (n) + "-1))", n, 2, 10)}))
Zip(Element(L, 2), L, IterationList({Element(lst, 1) + 1, ((2+cos(Element(lst, 1) + 1))/(sqrt(Element(lst, 1) + 1))) * Element(lst, 2)}, lst, {{1,
1}}, 9))
Zip(Elemento(L, 2), L, ListaIteración({Elemento(lst, 1) + 1, ((2+cos(Elemento(lst, 1) + 1))/(sqrt(Elemento(lst, 1) + 1))) * Elemento(lst, 2)},
lst, {{1, 1}}, 9))
Dos parámetros de recursión sin la variable de iteración i
Execute[Join[{"f_{1} = 1", "f_{2}=1" }, Sequence["f_{"+(i+2)+"} = f_{"+(i+1)+"}+f_{"+i+"}",i, 1, 10]]]
Ejecuta[Encadena[{"f_{1} = 1", "f_{2}=1" }, Secuencia["f_{"+(i+2)+"} = f_{"+(i+1)+"}+f_{"+i+"}",i, 1, 10]]]
IterationList(f1 + f2, f1, f2, {1, 2}, 10)
ListaIteración(f1 + f2, f1, f2, {1, 2}, 10)
Un parámetro de recursión sin la variable de iteración i
Execute[Join[{"f_{1} =11"}, Sequence["f_{"+(i)+"}=f_{"+(i-1)+"}+10",i, 2, 10]]]
Ejecuta[Encadena[{"f_{1}=11"},Secuencia["f_{"+(i)+"}=f_{"+(i-1)+"}+10",i, 2, 10]]]
IterationList(f1+10, f1, {11}, 9)
ListaIteración(f1+10, f1, {11}, 9)
Secuencias Recursivas con Geogebra Prof. Pérez Sanabria

3. Execute[Join[{"f_{1} =-3"}, Sequence["f_{"+(i)+"}=f_{"+(i-1)+"}*3",i, 2, 10]]]
Ejecuta[Encadena[{"f_{1}=-3"},Secuencia["f_{"+(i)+"}=f_{"+(i-1)+"}*3",i, 2, 10]]]
IterationList(f1*3, f1, {-3}, 9)
ListaIteración(f1*3, f1, {-3}, 9)
Execute[Join[{"f_{1} =-14"}, Sequence["f_{"+(i)+"}=1+f_{"+(i-1)+"}*(1/2)",i, 2, 10]]]
Ejecuta[Encadena[{"f_{1}=-14"},Secuencia["f_{"+(i)+"}=1+f_{"+(i-1)+"}*(1/2)",i, 2, 10]]]
IterationList(1+(1/2)*f1, f1, {-14}, 9)
ListaIteración(1+(1/2)*f1, f1, {-14}, 9)
Execute[Join[{"f_{1} =15"}, Sequence["f_{"+(i)+"}=f_{"+(i-1)+"}+200",i, 2, 10]]]
Ejecuta[Encadena[{"f_{1}=15"},Secuencia["f_{"+(i)+"}=f_{"+(i-1)+"}+200",i, 2, 10]]]
IterationList(f1+200, f1, {15}, 9)
ListaIteración(f1+200, f1, {15}, 9)
Execute[Join[{"f_{1}=3"},Sequence["f_{"+(i)+"}=1/(4-f_{"+(i-1)+"})",i, 2, 10]]]
Ejecuta[Encadena[{"f_{1}=3"},Secuencia["f_{"+(i)+"}=1/(4-f_{"+(i-1)+"})",i, 2, 10]]]
IterationList(1/(4-f1), f1, {3}, 9)
ListaIteración(1/(4-f1), f1, {3}, 9)
Execute[Join[{"f_{0}=4"},Sequence["f_{"+(i)+"}=(1/2)*f_{"+(i-1)+"}+4",i,1, 10]]]
Ejecuta[Encadena[{"f_{0}=4"},Secuencia["f_{"+(i)+"}=(1/2)*f_{"+(i-1)+"}+4",i,1, 10]]]
IterationList((1/2)*f1+4, f1, {6}, 9)
ListaIteración((1/2)*f1+4, f1, {6}, 9)
Execute[Join[{"f_{0}=3/2"},Sequence["f_{"+(i)+"}=f_{"+(i-1)+"}^2-2*f_{"+(i-1)+"}+2",i,1, 10]]]
Ejecuta[Encadena[{"f_{0}=3/2"},Secuencia["f_{"+(i)+"}=f_{"+(i-1)+"}^2-2*f_{"+(i-1)+"}+2",i,1, 10]]]
IterationList(f1^2-2*f1+2, f1, {5/4}, 9)
ListaIteración(f1^2-2*f1+2, f1, {5/4}, 9)
Secuencias Recursivas con Geogebra Prof. Pérez Sanabria

4. Execute[Join[{"f_{0}=11"},Sequence["f_{"+(i)+"}=- sqrt(24-2*f_{"+(i-1)+"})",i,1, 10]]]
Ejecuta[Encadena[{"f_{0}=11"},Secuencia["f_{"+(i)+"}=- sqrt(24-2*f_{"+(i-1)+"})",i,1, 10]]]
IterationList(-sqrt(24-2*f1), f1, {-sqrt(2)}, 9)
ListaIteración(-sqrt(24-2*f1), f1, {-sqrt(2)}, 9)
Execute[Join[{"f_{0}=1"},Sequence["f_{"+(i)+"}=2*f_{"+(i-1)+"}^(2/3)",i,1, 10]]]
Ejecuta[Encadena[{"f_{0}=1"},Secuencia["f_{"+(i)+"}=2*f_{"+(i-1)+"}^(2/3)",i,1, 10]]]
IterationList(2*f1^(2/3), f1, {2}, 9)
ListaIteración(2*f1^(2/3), f1, {2}, 9)
Execute[Join[{"f_{0}=5"},Sequence["f_{"+(i)+"}=2-1/(2+f_{"+(i-1)+"})",i,1, 10]]]
Ejecuta[Encadena[{"f_{0}=5"},Secuencia["f_{"+(i)+"}=2-1/(2+f_{"+(i-1)+"})",i,1, 10]]]
IterationList(2-1/(2+f1), f1, {13/7}, 9)
ListaIteración(2-1/(2+f1), f1, {13/7}, 9)
Execute[Join[{"f_{0}=1"},Sequence["f_{"+(i)+"}=(5*f_{"+(i-1)+"})/(3+f_{"+(i-1)+"})",i,1, 10]]]
Ejecuta[Encadena[{"f_{0}=1"},Secuencia["f_{"+(i)+"}=(5*f_{"+(i-1)+"})/(3+f_{"+(i-1)+"})",i,1, 10]]]
IterationList((5*f1)/(3+f1), f1, {5/4}, 9)
ListaIteración((5*f1)/(3+f1), f1, {5/4}, 9)
Execute[Join[{"f_{0}=15"},Sequence["f_{"+(i)+"}=(1/2)*(f_{"+(i-1)+"}+9/f_{"+(i-1)+"})",i,1, 10]]]
Ejecuta[Encadena[{"f_{0}=15"},Secuencia["f_{"+(i)+"}=(1/2)*(f_{"+(i-1)+"}+9/f_{"+(i-1)+"})",i,1, 10]]]
IterationList((1/2)*(f1+9/f1), f1, {39/5}, 9)
ListaIteración((1/2)*(f1+9/f1), f1, {39/5}, 9)
Execute[Join[{"f_{0}=2"},Sequence["f_{"+(i)+"}=f_{"+(i-1)+"}/2+1/f_{"+(i-1)+"}",i,1, 10]]]
Ejecuta[Encadena[{"f_{0}=2"},Secuencia["f_{"+(i)+"}=f_{"+(i-1)+"}/2+1/f_{"+(i-1)+"}",i,1, 10]]]
IterationList(f1/2+1/f1, f1, {3/2}, 9)
ListaIteración(f1/2+1/f1, f1, {3/2}, 9)
Referencia
Ami, Raymond & José Pérez: How to create recursive sequences in list form. Retrieve of: https://help.geogebra.org/topic/how-to-create-
recursive-sequences-in-list-form#
Secuencias Recursivas con Geogebra Prof. Pérez Sanabria