Problem 3 - Calculus Early Transcendentals

Question

What is meant by the ratio of a geometric series?

Step-by-Step Solution

Answer

Answer: The ratio of a geometric series, denoted as 'r,' is the constant factor by which each term is multiplied to get to the next term. It indicates the relationship between consecutive terms in the series. To calculate the common ratio (r) in a given geometric series, follow these steps: 1. Choose any two consecutive terms in the series. 2. Divide the second term by the first term. 3. The result obtained is the common ratio (r) of the geometric series.

Step 1: Introduction to Geometric Series

A geometric series is a mathematical sequence of numbers in which each term after the first one is determined by multiplying the previous term by a fixed, non-zero constant, called the common ratio (denoted as r).

Step 2: Formula of Geometric Series

To better understand the ratio in a geometric series, let's first look at the formula for a geometric series. A general geometric sequence can be represented as: a_n = a_1 * r^(n-1) where 'a_n' is the nth term in the series, 'a_1' is the first term, 'r' is the common ratio, and 'n' is the position of the term in the series.

Step 3: Ratio in a Geometric Series

The ratio (r) in a geometric series is the constant factor by which each term is multiplied to get to the next term. It indicates the relationship between consecutive terms in the series. If r > 1, the series increases, and if 0 < r < 1, the series decreases. For example, consider the geometric series: 2, 10, 50, 250, ... To find the ratio, you can divide any term by the previous term: r = a_2 / a_1 = a_3 / a_2 = a_4 / a_3 = ... In this example: r = 10/2 = 50/10 = 250/50 = 5 This means that each term in this geometric series is multiplied by 5 to get the next term.

Step 4: Calculating the Ratio in a Geometric Series

To calculate the common ratio (r) in a given geometric series, follow these steps: 1. Choose any two consecutive terms in the given series. 2. Divide the second term by the first term. 3. The result obtained is the common ratio (r) of the geometric series. For example, if the given geometric series is: 3, 6, 12, 24, ... 1. Choose any two consecutive terms, such as 6 and 12. 2. Divide the second term (12) by the first term (6): 12/6 = 2. 3. The common ratio (r) for this geometric series is 2. In this way, you can find the ratio of any geometric series.