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Magazine Chapter 14: Problem 19
Find the sum of the terms of each infinite geometric sequence. $$2,-\frac{1}{4}, \frac{1}{32}, \ldots$$
Short Answer
Expert verified
The sum of the infinite geometric sequence is \( \frac{16}{9} \).
Step by step solution
01
Identify the first term
In a geometric sequence, the first term is denoted by \( a \). In the given sequence \( 2, -\frac{1}{4}, \frac{1}{32}, \ldots \), the first term \( a = 2 \).
02
Determine the common ratio
The common ratio \( r \) can be found by dividing the second term by the first term, or any term by the previous term. Here, \( r = \frac{-\frac{1}{4}}{2} = -\frac{1}{8} \).
03
Verify the common ratio
To ensure accuracy, verify by calculating the ratio between other terms: \( \frac{\frac{1}{32}}{-\frac{1}{4}} = -\frac{1}{8} \). The sequence consistently maintains the common ratio of \( -\frac{1}{8} \).
04
Apply the sum formula for an infinite geometric series
The sum of an infinite geometric series is given by the formula \( S = \frac{a}{1-r} \), where \( |r| < 1 \). Here, \( r = -\frac{1}{8} \), so \( -1 < -\frac{1}{8} < 1 \), hence the sum exists.
05
Calculate the sum
Substitute the values into the sum formula: \( S = \frac{2}{1 - (-\frac{1}{8})} = \frac{2}{1 + \frac{1}{8}} = \frac{2}{\frac{9}{8}} = 2 \times \frac{8}{9} = \frac{16}{9} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Geometric Sequence
A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the 'common ratio'. Geometric sequences can appear as infinite, meaning they go on forever, or finite, where they have a limited set of terms.
Example: The sequence provided in the exercise, which is \( 2, -\frac{1}{4}, \frac{1}{32}, \ldots \), is an infinite geometric sequence. It begins with a starting value of 2, and each subsequent number is the product of the previous term and the common ratio.
- In a geometric sequence:
Example: The sequence provided in the exercise, which is \( 2, -\frac{1}{4}, \frac{1}{32}, \ldots \), is an infinite geometric sequence. It begins with a starting value of 2, and each subsequent number is the product of the previous term and the common ratio.
- In a geometric sequence:
- Each term is influenced by the one before it.
- The size of the common ratio determines how quickly the terms increase or decrease.
Common Ratio
In a geometric sequence, the common ratio is the constant factor that you multiply by each term to get the next term. To find the common ratio \( r \), you take any term and divide it by the preceding term.
In the original exercise, we found the common ratio by dividing the second term \( -\frac{1}{4} \) by the first term \( 2 \). This results in \( r = -\frac{1}{8} \).
Verifying with other terms, like dividing the third term \( \frac{1}{32} \) by the second \( -\frac{1}{4} \), also results in \( -\frac{1}{8} \). This consistency confirms that the sequence has a valid geometric progression.
In the original exercise, we found the common ratio by dividing the second term \( -\frac{1}{4} \) by the first term \( 2 \). This results in \( r = -\frac{1}{8} \).
Verifying with other terms, like dividing the third term \( \frac{1}{32} \) by the second \( -\frac{1}{4} \), also results in \( -\frac{1}{8} \). This consistency confirms that the sequence has a valid geometric progression.
- It is crucial that the common ratio be less than 1 in absolute value (\( |r| < 1 \)) for the infinite series to have a sum.
- If \( r > 1 \) or \( r < -1 \), the series does not converge.
Sum Formula
Calculating the sum of an infinite geometric series is straightforward with the sum formula. This formula, \( S = \frac{a}{1 - r} \), grants us a simple method to find the sum as long as the common ratio's absolute value is less than one.
In our exercise's case, with a first term \( a = 2 \) and a common ratio \( r = -\frac{1}{8} \), the series converges because \( |r| < 1 \).
Here's how it works:
In our exercise's case, with a first term \( a = 2 \) and a common ratio \( r = -\frac{1}{8} \), the series converges because \( |r| < 1 \).
Here's how it works:
- Plug the first term \( a \) and the common ratio \( r \) into the formula.
- Solve for the sum \( S \). In this instance, it's \( S = \frac{2}{1 - (-\frac{1}{8})} = \frac{2}{\frac{9}{8}} = \frac{16}{9} \).
First Term
The first term of a geometric sequence is known as the term from which the sequence begins. This initial value \( a \) is pivotal because each subsequent term is a progression based on it.
For example, in the sequence \( 2, -\frac{1}{4}, \frac{1}{32}, \ldots \), the first term \( a \) is 2. This starting point influences how the entire sequence unfolds.
- In any sequence:
For example, in the sequence \( 2, -\frac{1}{4}, \frac{1}{32}, \ldots \), the first term \( a \) is 2. This starting point influences how the entire sequence unfolds.
- In any sequence:
- The first term provides the initial jump-off value from which the series develops.
- It's essential for establishing the sequence pattern's root, as every following term multiplies off this starting number.
