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Para identificar cuál de las secuencias dadas es una progresión geométrica, debemos comprobar si la razón entre términos consecutivos es constante en cada una de las secuencias.
1. Primera secuencia: [tex]\(\frac{1}{8}, \frac{3}{4}, 1, \frac{15}{2}, \ldots\)[/tex]
- [tex]\(\frac{\frac{3}{4}}{\frac{1}{8}} = \frac{3}{4} \times \frac{8}{1} = 6\)[/tex]
- [tex]\(\frac{1}{\frac{3}{4}} = \frac{4}{3}\)[/tex]
- [tex]\(\frac{\frac{15}{2}}{1} = \frac{15}{2}\)[/tex]
La razón no es constante.
2. Segunda secuencia: [tex]\(\frac{2}{3}, \frac{11}{12}, \frac{7}{6}, \frac{17}{12}, \cdots\)[/tex]
- [tex]\(\frac{\frac{11}{12}}{\frac{2}{3}} = \frac{11}{12} \times \frac{3}{2} = \frac{33}{24} = \frac{11}{8}\)[/tex]
- [tex]\(\frac{\frac{7}{6}}{\frac{11}{12}} = \frac{7}{6} \times \frac{12}{11} = \frac{84}{66} = \frac{14}{11}\)[/tex]
- [tex]\(\frac{\frac{17}{12}}{\frac{7}{6}} = \frac{17}{12} \times \frac{6}{7} = \frac{102}{84} = \frac{17}{14}\)[/tex]
La razón no es constante.
3. Tercera secuencia: [tex]\(\frac{2}{5}, \frac{2}{25}, \frac{6}{125}, \frac{24}{625}, \cdots\)[/tex]
- [tex]\(\frac{\frac{2}{25}}{\frac{2}{5}} = \frac{2}{25} \times \frac{5}{2} = \frac{10}{50} = \frac{1}{5}\)[/tex]
- [tex]\(\frac{\frac{6}{125}}{\frac{2}{25}} = \frac{6}{125} \times \frac{25}{2} = \frac{150}{250} = \frac{3}{5}\)[/tex]
- [tex]\(\frac{\frac{24}{625}}{\frac{6}{125}} = \frac{24}{625} \times \frac{125}{6} = \frac{3000}{3750} = \frac{4}{5}\)[/tex]
La razón no es constante.
4. Cuarta secuencia: [tex]\(\frac{1}{4}, \frac{3}{2}, 9, 54, \cdots\)[/tex]
- [tex]\(\frac{\frac{3}{2}}{\frac{1}{4}} = \frac{3}{2} \times \frac{4}{1} = 6\)[/tex]
- [tex]\(\frac{9}{\frac{3}{2}} = 9 \times \frac{2}{3} = 6\)[/tex]
- [tex]\(\frac{54}{9} = 6\)[/tex]
La razón es constante.
Por lo tanto, la cuarta secuencia ([tex]\(\frac{1}{4}, \frac{3}{2}, 9, 54, \cdots\)[/tex]) es una progresión geométrica.
1. Primera secuencia: [tex]\(\frac{1}{8}, \frac{3}{4}, 1, \frac{15}{2}, \ldots\)[/tex]
- [tex]\(\frac{\frac{3}{4}}{\frac{1}{8}} = \frac{3}{4} \times \frac{8}{1} = 6\)[/tex]
- [tex]\(\frac{1}{\frac{3}{4}} = \frac{4}{3}\)[/tex]
- [tex]\(\frac{\frac{15}{2}}{1} = \frac{15}{2}\)[/tex]
La razón no es constante.
2. Segunda secuencia: [tex]\(\frac{2}{3}, \frac{11}{12}, \frac{7}{6}, \frac{17}{12}, \cdots\)[/tex]
- [tex]\(\frac{\frac{11}{12}}{\frac{2}{3}} = \frac{11}{12} \times \frac{3}{2} = \frac{33}{24} = \frac{11}{8}\)[/tex]
- [tex]\(\frac{\frac{7}{6}}{\frac{11}{12}} = \frac{7}{6} \times \frac{12}{11} = \frac{84}{66} = \frac{14}{11}\)[/tex]
- [tex]\(\frac{\frac{17}{12}}{\frac{7}{6}} = \frac{17}{12} \times \frac{6}{7} = \frac{102}{84} = \frac{17}{14}\)[/tex]
La razón no es constante.
3. Tercera secuencia: [tex]\(\frac{2}{5}, \frac{2}{25}, \frac{6}{125}, \frac{24}{625}, \cdots\)[/tex]
- [tex]\(\frac{\frac{2}{25}}{\frac{2}{5}} = \frac{2}{25} \times \frac{5}{2} = \frac{10}{50} = \frac{1}{5}\)[/tex]
- [tex]\(\frac{\frac{6}{125}}{\frac{2}{25}} = \frac{6}{125} \times \frac{25}{2} = \frac{150}{250} = \frac{3}{5}\)[/tex]
- [tex]\(\frac{\frac{24}{625}}{\frac{6}{125}} = \frac{24}{625} \times \frac{125}{6} = \frac{3000}{3750} = \frac{4}{5}\)[/tex]
La razón no es constante.
4. Cuarta secuencia: [tex]\(\frac{1}{4}, \frac{3}{2}, 9, 54, \cdots\)[/tex]
- [tex]\(\frac{\frac{3}{2}}{\frac{1}{4}} = \frac{3}{2} \times \frac{4}{1} = 6\)[/tex]
- [tex]\(\frac{9}{\frac{3}{2}} = 9 \times \frac{2}{3} = 6\)[/tex]
- [tex]\(\frac{54}{9} = 6\)[/tex]
La razón es constante.
Por lo tanto, la cuarta secuencia ([tex]\(\frac{1}{4}, \frac{3}{2}, 9, 54, \cdots\)[/tex]) es una progresión geométrica.
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