Problem 45 - A Graphical Approach to Precalculus with Limits

Question

Find the sum for each series. $$\sum_{i=1}^{5}\left(3^{i}-4\right)$$

Step-by-Step Solution

Answer

The sum of the series is 343.

Step 1: Understanding the Problem

We are given a series where we need to find the sum: \(\sum_{i=1}^{5}\left(3^{i}-4\right)\). This means we need to calculate \(3^{i} - 4\) for each \(i\) from 1 to 5 and then add those results together.

Step 2: Evaluating Terms in the Series

Let's evaluate each term in the series separately:- For \(i=1\), the term is \(3^{1} - 4 = 3 - 4 = -1\).- For \(i=2\), the term is \(3^{2} - 4 = 9 - 4 = 5\).- For \(i=3\), the term is \(3^{3} - 4 = 27 - 4 = 23\).- For \(i=4\), the term is \(3^{4} - 4 = 81 - 4 = 77\).- For \(i=5\), the term is \(3^{5} - 4 = 243 - 4 = 239\).

Step 3: Calculating the Sum

Now that we have each term:- Add the terms: \(-1 + 5 + 23 + 77 + 239\).- Perform the addition step by step: - \(-1 + 5 = 4\) - \(4 + 23 = 27\) - \(27 + 77 = 104\) - \(104 + 239 = 343\).

Step 4: Finalizing the Solution

The sum of the series \(\sum_{i=1}^{5}(3^{i} - 4)\) is \(343\). We calculated each term and then added them together to arrive at the final sum.

Key Concepts

Sum of Powers

When we refer to the "sum of powers" in mathematics, it typically involves calculating the summation of exponential expressions. In this exercise, we specifically worked with the expression \(3^i\), where \(i\) indicates the power to which 3 is raised.

The task involves computing these powers systematically as part of a series from \(i=1\) to \(i=5\). Here, the exponential function grows very quickly; for instance, \(3^1 = 3\), \(3^2 = 9\), and by the time we reach \(3^5\), the value skyrockets to 243. This is because the base number, 3, is multiplied by itself multiple times.

Understanding this growth is crucial, as it demonstrates how rapidly numbers escalate when dealing with powers. Such exponential series are common in various areas of mathematics including sequences and calculus, offering critical insights into growth patterns.

Arithmetic Operations

Arithmetic operations are the fundamental procedures in mathematics, such as addition, subtraction, multiplication, and division. In solving a series problem like this, we applied several arithmetic operations.

For every term in the series \(3^i - 4\), subtraction is initially used to adjust the power of 3 by removing 4. This results in a new list of adjusted values. Next, addition is employed to sum these evaluated terms: \(-1, 5, 23, 77, 239\).

Here’s a breakdown of how these operations work together:

• Subtraction: Discards a constant value from each power of 3.
• Addition: Compiles all adjusted values to find the final sum of the series.

This integration of operations builds the foundational arithmetic skills often used in algebra and calculus to address more complex mathematical problems.

Series Evaluation

Evaluating a series, like in the given exercise, involves calculating each component systematically and then finding their cumulative total. Series evaluation is an important skill that requires both logical processing and accuracy.

The problem provided involves the series \(\sum_{i=1}^{5}(3^i - 4)\). We began by evaluating each term separately:

• \(i=1:\; 3^1 - 4 = -1\)
• \(i=2:\; 3^2 - 4 = 5\)
• \(i=3:\; 3^3 - 4 = 23\)
• \(i=4:\; 3^4 - 4 = 77\)
• \(i=5:\; 3^5 - 4 = 239\)

Finally, these results are summed as \(-1 + 5 + 23 + 77 + 239 = 343\). This methodical approach ensures that each term is accurately calculated, illustrating the power of breaking down problems into smaller parts.

Learning to evaluate series is essential. It enables deeper understanding of sequences, helps solve mathematical patterns, and develops problem-solving skills that are applicable in advanced mathematics and various real-world scenarios.