Problem 26 - A Graphical Approach to College Algebra

Question

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 25 and 26 to the nearest hundredth. $$a_{1}=-3.772, r=-1.553$$

Step-by-Step Solution

Answer

The sum of the first five terms is approximately -18.35.

Step 1: Identify the Formula

For a geometric sequence, the sum of the first \( n \) terms is calculated using the formula for the sum of a geometric series: \( S_n = a_1 \frac{1-r^n}{1-r} \), where \( a_1 \) is the first term and \( r \) is the common ratio.

Step 2: Substitute Values

Substitute the given values into the formula. Here, \( a_1 = -3.772 \), \( r = -1.553 \), and \( n = 5 \). The formula becomes: \[ S_5 = -3.772 \frac{1-(-1.553)^5}{1-(-1.553)} \]

Step 3: Calculate \( r^n \)

Calculate \( (-1.553)^5 \). Use a calculator to find \( (-1.553)^5 = -11.414 \) approximately.

Step 4: Substitute \( r^n \) into Formula

Substitute \( r^n = -11.414 \) back into the formula: \[ S_5 = -3.772 \frac{1 - (-11.414)}{1 - (-1.553)} \] which simplifies to \[ S_5 = -3.772 \frac{1 + 11.414}{1 + 1.553} \]

Step 5: Calculate Denominator and Numerator

Calculate the denominator and numerator separately. The denominator \( 1 + 1.553 = 2.553 \), and the numerator \( 1 + 11.414 = 12.414 \).

Step 6: Calculate the Fraction

Compute the fraction \( \frac{12.414}{2.553} \). Using a calculator, this is approximately equal to \( 4.864 \).

Step 7: Multiply by \( a_1 \)

Multiply the result by \( a_1 = -3.772 \): \( -3.772 \times 4.864 \approx -18.353 \).

Step 8: Round the Result

Round the final result to the nearest hundredth: \( -18.353 \approx -18.35 \).

Key Concepts

Sum of Geometric Series

The sum of a geometric series is a fascinating concept in mathematics, involving the addition of a set of terms that come from a geometric sequence. To find this sum, we use a specific formula: \( S_n = a_1 \frac{1-r^n}{1-r} \), where \( S_n \) denotes the sum of the first \( n \) terms. Here, \( a_1 \) is the initial term of the sequence, \( r \) is the common ratio, and \( n \) is the number of terms.

The formula assumes that \( r \), the common ratio, is not equal to 1. When \( r = 1 \), all terms are identical, making the sum simply \( a_1 \times n \). By leveraging this formula, you can calculate the sum of any finite geometric sequence efficiently.

• Numerator: The expression \( 1-r^n \) in the formula captures the effect of the common ratio on the sequence's overall growth or decline.
• Denominator: The expression \( 1-r \) ensures that the changing terms are adjusted by how much they deviate from 'staying the same'.

By substituting known values into the formula, as shown in the solution, you can find the sum of specified terms in a sequence with precision.

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is obtained by multiplying the previous one by a fixed number known as the common ratio. Let's take an example to delve deeper into understanding.

In the original exercise, the given sequence begins with \( a_1 = -3.772 \), and continues such that each term is a result of multiplying the previous term by the common ratio \( r = -1.553 \). This transformation yields a pattern characterized by alternating signs and compounding growth or reduction due to the absolute value of the ratio being bigger than one.

• Term Generation: Every new number springs from its predecessor by multiplying the common ratio.
• Sequence Traits: The sign of \( r \) dictates whether the terms switch between positive and negative.
• Magnitude: The magnitude of \( r \) determines whether the sequence grows, diminishes, or remains constant in absolute terms.

By following these principles, we see how geometric sequences form the basis for computing their sum systematically.

Common Ratio

The common ratio is a core idea in mastering geometric sequences and series. It signifies the constant factor that links consecutive terms in a sequence. If you know the common ratio, predicting the sequence's behavior becomes manageable.

In our example, \( r = -1.553 \). This negative ratio causes terms to swap signs successively, creating an oscillating series that can shrink or grow depending on whether \(|r| > 1\) or \(|r| < 1\).

• Positive vs Negative: A positive \( r \) keeps all terms' signs uniform, while a negative \( r \) creates alternation.
• Growth: When \(|r| > 1\), the sequence 'expands' as each step multiplies the term's magnitude.
• Contraction: When \(|r| < 1\), the sequence 'contracts', potentially converging towards zero.

Understanding the common ratio lets us predict a sequence's structure and compute related sums, as required in various mathematical and real-world applications.