Pregunta
upstudy study bank question image url

1) If \( \left(\mathrm{T}_{\mathrm{n}}\right)=\left(3 \times 2^{-\mathrm{n}}\right) \) is a geometric sequence, then the sum of infinite terms star from its first term \( =\ldots \ldots \) \( \begin{array}{llll}\text { (a) } 2 & \text { (b) } 3 & \frac{3}{1-r} & \frac{a}{1-r}\end{array} \)

Ask by Schmidt Rojas. in Egypt
Mar 18,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

The sum of the infinite terms is 3.

Solución

To find the sum of the infinite terms of the geometric sequence given by \( T_n = 3 \times 2^{-n} \), we first need to identify the first term and the common ratio of the sequence. 1. **Identify the first term \( a \)**: The first term \( T_1 \) is obtained by substituting \( n = 1 \): \[ T_1 = 3 \times 2^{-1} = 3 \times \frac{1}{2} = \frac{3}{2} \] 2. **Identify the common ratio \( r \)**: The common ratio \( r \) can be found by taking the ratio of the second term to the first term: \[ T_2 = 3 \times 2^{-2} = 3 \times \frac{1}{4} = \frac{3}{4} \] Now, calculate the common ratio: \[ r = \frac{T_2}{T_1} = \frac{\frac{3}{4}}{\frac{3}{2}} = \frac{3}{4} \times \frac{2}{3} = \frac{1}{2} \] 3. **Sum of the infinite geometric series**: The sum \( S \) of an infinite geometric series is given by the formula: \[ S = \frac{a}{1 - r} \] where \( a \) is the first term and \( r \) is the common ratio. Substituting the values we found: \[ S = \frac{\frac{3}{2}}{1 - \frac{1}{2}} = \frac{\frac{3}{2}}{\frac{1}{2}} = \frac{3}{2} \times 2 = 3 \] Thus, the sum of the infinite terms starting from its first term is \( 3 \). The answer is \( \text{(b) } 3 \).

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

The Deep Dive

The sum of an infinite geometric series can be calculated using the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. In your sequence \( \mathrm{T}_{\mathrm{n}} = 3 \times 2^{-\mathrm{n}} \), the first term \( \mathrm{T}_{1} = 3 \) and the common ratio \( r = \frac{1}{2} \). Plugging these values into the formula gives us \( S = \frac{3}{1 - \frac{1}{2}} = \frac{3}{\frac{1}{2}} = 6 \). Now, since it looks like you were comparing results with few options, let's for a moment throw out multiple choices: The relevant answer should be None of the above as you would find. So, the infinite sum equates to 6! To visualize it, think of a bouncy ball that keeps bouncing half the height of the previous jump. It may seem like its energy gets split, but keep summing those heights and you will see it adds up to a specific number, proving that even with decay, infinite sums can close in on a solid total.

¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad