Problem 13 - Algebra 2

Question

Find the sum of each infinite geometric series, if it exists. \(a_{1}=4, r=\frac{5}{7}\)

Step-by-Step Solution

Answer

The sum of the series is 14.

Step 1: Identify the Formula

The sum of an infinite geometric series can be found using the formula \( S = \frac{a_1}{1 - r} \) where \( a_1 \) is the first term and \( r \) is the common ratio. This formula is applicable when \( |r| < 1 \).

Step 2: Verify the Condition

Check if the common ratio \( r = \frac{5}{7} \) satisfies \( |r| < 1 \). Since \( \left|\frac{5}{7}\right| = \frac{5}{7} \approx 0.714 < 1 \), the condition is met, and the series sum can be calculated.

Step 3: Substitute Values

Substitute \( a_1 = 4 \) and \( r = \frac{5}{7} \) into the formula: \[ S = \frac{4}{1 - \frac{5}{7}} \]

Step 4: Perform the Calculation

First, calculate the denominator:\( 1 - \frac{5}{7} = \frac{7}{7} - \frac{5}{7} = \frac{2}{7} \).Next, substitute back into the sum formula:\[ S = \frac{4}{\frac{2}{7}} \]Simplify by multiplying by the reciprocal:\[ S = 4 \times \frac{7}{2} = \frac{28}{2} = 14 \]

Step 5: Verify the Result

Ensure all calculations were performed correctly. The substitution and simplification steps confirm the series sum is \( 14 \).

Key Concepts

sum of geometric series

Understanding the sum of a geometric series is crucial in mathematics, especially when dealing with sequences that expand infinitely. The sum of an infinite geometric series can be calculated using the formula:

• \( S = \frac{a_1}{1 - r} \)

where \( a_1 \) is the first term of the series, and \( r \) is the common ratio. This formula allows us to find a finite sum, even if the sequence itself extends infinitely. When the common ratio \(|r|\) is less than 1, the terms of the sequence get closer and closer to zero, which is why the series converges to a specific sum, \( S \). This formula is particularly powerful because it provides a straightforward method to calculate what might at first seem an impossibly large summation.

common ratio

The common ratio in a geometric series is a vital component. It determines how the consecutive terms in the series multiply from one term to the next. For a series denoted as \( a_1, a_2, a_3, \ldots \), the common ratio \( r \) derives from the equation

• \( a_{n+1} = r \cdot a_n \)

Knowing the common ratio tells us how the series grows or shrinks. For instance, if \( r > 1 \), the series grows exponentially, while if \( 0 < r < 1 \), the series gradually decreases, converging towards zero. It's this property that allows series with \(|r| < 1\) to have a calculable infinite sum. In our solution, the common ratio \( r = \frac{5}{7} \) met the \(|r| < 1\) condition, enabling us to find the series' sum effectively.

infinite series convergence

Convergence is a concept that describes the behavior of an infinite series as the number of terms grows indefinitely. For a geometric series to converge, the absolute value of the common ratio must be less than 1, i.e.,

• \(|r| < 1 \)

This condition ensures that as we continue adding terms, the subsequent terms become increasingly smaller and their collective sum approaches a fixed point. When dealing with an infinite geometric series, convergence signifies that the sum is not infinite; rather, it sums up to a definite number. This principle underlines the calculation of the infinite geometric series sum using

• \( S = \frac{a_1}{1 - r} \)

thereby providing clarity and a finite result, such as \( 14 \) in the problem we analyzed. Understanding convergence helps demystify why and how infinite series can yield meaningful and tangible outcomes.