Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Find the sum of the infinite geometric series if it exists. $$256+192+144+108+\cdots$$

Step-by-Step Solution

Answer

The sum of the infinite geometric series is 1024.

Step 1: Identify the first term

The first term of the geometric series is the first number in the sequence. From the series given, the first term \( a = 256 \).

Step 2: Find the common ratio

The common ratio \( r \) is found by dividing the second term by the first term. In this case, \( r = \frac{192}{256} = \frac{3}{4} \).

Step 3: Determine if an infinite sum exists

An infinite geometric series has a sum if the absolute value of the common ratio is less than 1. Since \( |r| = \left| \frac{3}{4} \right| = \frac{3}{4} \) which is less than 1, the sum exists.

Step 4: Calculate the sum of the series

The sum \( S \) of an infinite geometric series with \( |r| < 1 \) is given by the formula \( S = \frac{a}{1 - r} \). Substituting \( a = 256 \) and \( r = \frac{3}{4} \), we have: \[ S = \frac{256}{1 - \frac{3}{4}} = \frac{256}{\frac{1}{4}} \]Simplify the expression: \[ S = 256 \times 4 = 1024 \]

Step 5: Conclusion

The sum of the infinite geometric series is \( 1024 \).

Key Concepts

Understanding the Common Ratio

The concept of the common ratio is central to solving problems involving geometric series. In such series, each term after the first is multiplied by a fixed, non-zero number to arrive at the next term. This number is known as the common ratio, denoted generally by \( r \). To find the common ratio, you simply divide any term in the series by the previous term. For instance, in our exercise, the series starts with 256 and the next term is 192. By dividing 192 by 256, we compute the common ratio: \( r = \frac{192}{256} = \frac{3}{4} \). This ratio should remain consistent throughout a geometric series, indicating a pattern of multiplication. Recognizing and confirming the common ratio allows us to explore more about the behavior of the series, such as its convergence and potential sum.

Summing Up the Series

Finding the sum of an infinite geometric series involves understanding its pattern through the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. This formula only applies when the series converges, which hinges on the common ratio, \( r \).In our problem, the series starts at 256. Using the common ratio \( r = \frac{3}{4} \), which is less than 1, the series does indeed converge, allowing us to calculate the sum. Substituting into the formula: - First term \( a = 256 \)- Common ratio \( r = \frac{3}{4} \)We find the sum:\[S = \frac{256}{1 - \frac{3}{4}} = \frac{256}{\frac{1}{4}} = 256 \times 4 = 1024\]Hence, the sum of this infinite geometric series is 1024, a remarkable result showing that even an infinitely long series can converge to a finite value.

Convergence Criterion Demystified

The convergence criterion is a crucial principle in determining whether an infinite geometric series has a finite sum. This criterion states that the series converges if the absolute value of the common ratio \( |r| \) is less than 1. This means that as you continue to add more terms in the series, they contribute increasingly smaller values. When \( |r| \) is less than 1, the series terms diminish toward zero, ensuring that the sum approaches a finite limit.In our exercise, the common ratio \( r = \frac{3}{4} \) fits this criterion perfectly, since \( |\frac{3}{4}| = \frac{3}{4} < 1 \). This assures us that the series converges, allowing us to use the formula to find its sum, thereby confirming the real-world application of theoretical math concepts. Understanding this criterion helps in confidently solving problems involving infinite geometric series.