Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ a_{1}=4, r=3, n=11 $$

Step-by-Step Solution

Answer

The sum is 354292.

Step 1: Understand the Formula for Sum of Geometric Series

To find the sum of the first \( n \) terms of a geometric series, we use the formula: \[ S_n = a_1 \frac{1-r^n}{1-r} \] if \( r eq 1 \). In our case, the first term \( a_1 = 4 \), the common ratio \( r = 3 \), and the number of terms \( n = 11 \).

Step 2: Calculate \( r^n \)

Substitute \( r = 3 \) and \( n = 11 \) into the term \( r^n \) in the formula. Calculate this term: \[ r^{11} = 3^{11} = 177147 \].

Step 3: Substitute Values into the Formula

Plug \( a_1 = 4 \), \( r = 3 \), and \( r^n = 177147 \) into the sum formula: \[ S_{11} = 4 \frac{1-177147}{1-3} \].

Step 4: Simplify the Formula

Calculate the expression: \[ 1 - 177147 = -177146 \] and \(1 - 3 = -2\). Substitute back into the sum formula: \[ S_{11} = 4 \frac{-177146}{-2} \].

Step 5: Final Calculation

Complete the calculation: \[ S_{11} = 4 \times 88573 = 354292 \].

Step 6: Conclusion of the Sum

The sum of the first 11 terms of the geometric series is 354292.

Key Concepts

Sum of Geometric Series

In mathematics, a geometric series is a series of numbers where each term is the product of the previous term and a constant called the 'common ratio'. Calculating the sum of a geometric series involves using a specific formula. For any geometric series, the sum of the first \( n \) terms is given by:\[ S_n = a_1 \frac{1-r^n}{1-r} \]where:

• \( S_n \) is the sum of the series.
• \( a_1 \) represents the first term in the series.
• \( r \) stands for the common ratio.
• \( n \) is the number of terms.

This formula works when the common ratio \( r eq 1 \). It helps to find the cumulative total of terms quickly instead of adding each term manually. For instance, in the exercise, using the formula with \( a_1 = 4 \), \( r = 3 \), and \( n = 11 \), you find that the sum is 354292.

Common Ratio

The common ratio is a fundamental concept in geometric sequences and series. It is the constant factor you multiply by to get from one term to the next. In a geometric sequence, each term after the first is found by multiplying the previous term by this ratio.In the formula for the sum of a geometric series, the common ratio \( r \) appears in both the numerator and the denominator. It significantly affects the growth rate of the series:

• If \( |r| > 1 \), the terms will increase or decrease exponentially.
• If \( |r| < 1 \), the terms will converge toward zero.
• If \( r = 1 \), each term is equal to the first term, creating an arithmetic sequence.

In the given problem, the common ratio \( r \) is 3, meaning each term is three times the previous one, leading to rapid growth in the series.

Geometric Sequence

A geometric sequence is a set of numbers where each term after the first is obtained by multiplying the previous one by a fixed, non-zero number known as the 'common ratio'. This type of sequence has exponential characteristics due to the repeated multiplication by the constant ratio.A geometric sequence can be represented as:

• \( a_1 \)
• \( a_1 \times r \)
• \( a_1 \times r^2 \)
• \( a_1 \times r^3 \), and so on.

Each set maintains a consistent multiplication pattern throughout. Understanding this sequence helps to apply the sum formula correctly to get the desired total. In the exercise, starting with \( a_1 = 4 \) and \( r = 3 \), the sequence grows rapidly across its terms.

Number of Terms in a Series

The number of terms, denoted as \( n \), in a geometric series is crucial in determining the series' sum. It tells us how many terms are included from the first term onward in the calculation. The sum formula, \( S_n = a_1 \frac{1-r^n}{1-r} \), needs \( n \) to accurately compute the total, especially when the series has rapidly changing terms due to a high common ratio.To find \( n \), simply count all terms, beginning with the first term up to the term you want to include in your sum:

• If \( n = 1 \), only the first term is included.
• If \( n = 11 \), as in the exercise, all terms up to the 11th are included.

Understanding \( n \) ensures accurate application of the sum formula. It's essential as it influences how large \( r^n \) becomes in the calculations, impacting the overall sum significantly.