Problem 50 - College Algebra with Modeling and Visualization

Question

Find the sum of the infinite geometric series. $$ 25-5+1-\frac{1}{5}+\dots+25\left(-\frac{1}{5}\right)^{n-1}+\cdots $$

Step-by-Step Solution

Answer

The sum of the infinite geometric series is \( \frac{125}{6} \).

Step 1: Identify the First Term (a)

The first term of the series is given as 25. Therefore, \( a = 25 \).

Step 2: Determine the Common Ratio (r)

To find the common ratio, divide the second term by the first term: \( r = \frac{-5}{25} = -\frac{1}{5} \).

Step 3: Verify it is an Infinite Geometric Series

An infinite geometric series can be summed if the absolute value of the common ratio is less than 1: \( |r| = |-\frac{1}{5}| = \frac{1}{5} < 1 \).

Step 4: Use the Formula for Sum of Infinite Geometric Series

The formula for the sum of an infinite geometric series is \( S = \frac{a}{1-r} \) where \( a \) is the first term and \( r \) is the common ratio.

Step 5: Substitute Values into the Formula

Substitute \( a = 25 \) and \( r = -\frac{1}{5} \) into the formula: \[ S = \frac{25}{1 - \left(-\frac{1}{5}\right)} = \frac{25}{1 + \frac{1}{5}} = \frac{25}{\frac{6}{5}}. \]

Step 6: Simplify the Fraction

Simplify \( \frac{25}{\frac{6}{5}} \) by multiplying by the reciprocal: \[ S = 25 \times \frac{5}{6} = \frac{125}{6}. \]

Key Concepts

Sum of Series

In mathematics, the sum of an infinite geometric series can only be found if the series converges. A series converges when the absolute value of the common ratio is less than one. When working with infinite geometric series, the sum can be calculated using the formula:\[ S = \frac{a}{1-r} \]Where:

• \( S \) is the sum of the series.
• \( a \) represents the first term.
• \( r \) is the common ratio.

For instance, in our series, by substituting the first term (25) and the common ratio (-1/5) into the formula, we calculate the sum:\[ S = \frac{25}{1 - \left(-\frac{1}{5}\right)} = \frac{25}{\frac{6}{5}} \]This simplifies to:\[ S = \frac{125}{6} \]It's important to be familiar with this kind of formula to solve similar series problems quickly.

Common Ratio

The common ratio is a key part of understanding geometric sequences and series. It tells us how each term is related to its previous term. In a geometric series, each term after the first is found by multiplying the previous term by the common ratio. Therefore, recognizing the common ratio of a sequence is essential in identifying it as a geometric sequence.
The formula to find the common ratio \( r \) is to divide any term in the sequence by the preceding term:\[ r = \frac{a_2}{a_1} \]In our example, the second term is -5 and the first term is 25. Calculating the common ratio:\[ r = \frac{-5}{25} = -\frac{1}{5} \]The sign in the common ratio can significantly affect the behavior of the series, especially impacting whether the terms alternate between positive and negative values.

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous one by a fixed, non-zero number called the common ratio. Geometric sequences are identified by this constant factor that differentiates them from other sequences such as arithmetic sequences, where terms change by a constant sum.
In the geometric sequence from the problem, the sequence is 25, -5, 1, \(-\frac{1}{5}\), and continues indefinitely. Here, each term is derived by multiplying the previous term by \(-\frac{1}{5}\). This consistent multiplication confirms the sequence's nature as geometric, allowing us to subsequently leverage this feature to find formulas and solutions such as the sum for specific conditions like convergence.

First Term

The first term in a sequence or series is fundamental to building the rest of the sequence's terms. It is denoted as \( a \) in geometric sequences and series. All calculations related to the sequence, including finding subsequent terms or sums, rely heavily on this value.

• The first term of our series, as noted, is 25.
• Every term in the sequence is a factor of this initial term multiplied by powers of the common ratio.

Understanding the value of the first term can help set the stage for determining the behavior of the entire series. For this given problem, recognizing the first term and the formula's reliance on \( a \) was essential to effectively calculate the series sum as \( \frac{125}{6} \). Thus, the first term provides an anchor point for all following analysis and computations.