For the following, write its recursive definition: The image shows a mathematical expression related to products in sequences:\[\prod_{k=1}^{n} a_{k}, \text{ for } n \geq 1\]This expression represents the product of a sequence of terms $ a_k $, starting from $ k = 1 $ up to $ k = n $. The condition $ n \geq 1 $ indicates that the product is defined only for positive integer values of $ n $. Therefore, it calculates the multiplication of the terms $ a_1 \times a_2 \times \cdots \times a_n $. This notation is often used in mathematics to succinctly represent such multiplicative processes in sequences.

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Description: For the following, write its recursive definition: The image shows a mathematical expression related to products in sequences:\[\prod_{k=1}^{n} a_{k}, \text{ for } n \geq 1\]This expression represents the product of a sequence of terms $ a_k $, sta

Answered: For the following, write its recursive…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter9: Sequences, Probability And Counting Theory

Section9.1: Sequences And Their Notations

Problem 2SE: Describe three ways that a sequence can be defined.

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For the following, write its recursive definition:

The image shows a mathematical expression related to products in sequences:

\[
\prod_{k=1}^{n} a_{k}, \text{ for } n \geq 1
\]

This expression represents the product of a sequence of terms $ a_k $, starting from $ k = 1 $ up to $ k = n $. The condition $ n \geq 1 $ indicates that the product is defined only for positive integer values of $ n $. Therefore, it calculates the multiplication of the terms $ a_1 \times a_2 \times \cdots \times a_n $. This notation is often used in mathematics to succinctly represent such multiplicative processes in sequences.

Expert Solution

Step 1

Consider $\prod_{k = 1}^{n} a_{k}$ for $n \geq 1$ .

A recursive definition is one which defines the elements of the sequence in terms of the previous terms.

When $n = 1$ , we have

$\begin{array}{rcl} \prod_{k = 1}^{n} a_{k} & = & \prod_{k = 1}^{1} a_{k} \\ = & a_{1} \end{array}$

When $n = 2$ , we have

$\begin{array}{rcl} \prod_{k = 1}^{n} a_{k} & = & \prod_{k = 1}^{2} a_{k} \\ = & a_{1} a_{2} \end{array}$

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