Problem 49 - A Graphical Approach to Precalculus with Limits

Question

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\) $$\sum_{i=1}^{5}\left(2 x_{i}+3\right)$$

Step-by-Step Solution

Answer

The sum is 15.

Step 1: Understand the Expression

The given expression is a summation, \(\sum_{i=1}^{5}\left(2x_{i}+3\right)\), which means we need to evaluate it for each \(x_i\) from \(x_1\) to \(x_5\). We will be substituting each of these values into the expression \(2x_i + 3\) and then summing the results.

Step 2: Evaluate the Expression for Each Term

Substitute the given values one by one:- For \(x_1 = -2\): \(2(-2) + 3 = -4 + 3 = -1\)- For \(x_2 = -1\): \(2(-1) + 3 = -2 + 3 = 1\)- For \(x_3 = 0\): \(2(0) + 3 = 0 + 3 = 3\)- For \(x_4 = 1\): \(2(1) + 3 = 2 + 3 = 5\)- For \(x_5 = 2\): \(2(2) + 3 = 4 + 3 = 7\)

Step 3: Sum the Evaluated Terms

Add all the results obtained in Step 2:\(-1 + 1 + 3 + 5 + 7 = 15\).

Key Concepts

Understanding Series

A series in mathematics refers to the sum of a sequence of numbers. This is often denoted with the summation symbol \(\Sigma\), which indicates that you need to add up a list of terms. In the given exercise, the series consists of the summation of each term resulting from substituting values into the expression \(2x_i + 3\). When you see a problem involving a series, you're looking at an addition of several results from a pattern or formula applied over a range of numbers.

The key steps to solving a series involve:

• Identifying the function or expression that will produce each term of the series.
• Replacing variables with specific values to get those terms.
• Summing all the calculated terms to find the result of the series.

When calculating a series, make sure each term is evaluated individually before summing them all together. This systematic approach helps avoid mistakes and makes the process easier to follow.

Arithmetic Sequences: The Basics

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference. In the case of the exercise we're looking at, while the terms don't constitute a simple arithmetic sequence directly, understanding arithmetic sequences helps us recognize patterns in more complex series.

Let's break down the definition:

• First Term: The starting point of the sequence.
• Common Difference: The consistent change between each subsequent term.
• General Formula: Given as \(a_n = a_1 + (n-1) \cdot d\), where \(a_n\) is the nth term, \(a_1\) is the first term, and \(d\) is the common difference.

By applying this understanding, you can better anticipate the behavior of sequences and series within algebraic expressions.

The Role of Algebra in Summation

Algebra is fundamental in understanding and solving problems involving series and sequences. It provides the tools needed to manipulate equations and expressions to find unknowns and calculate totals like sums.

In the exercise, we're using algebraic manipulation to evaluate each term in the series. Here's how algebra plays a role:

• Expression Evaluation: Algebra allows us to substitute specific values for variables, supporting the calculation of individual terms, like \(2x_i + 3\).
• Simplification: It helps simplify complicated expressions, which is critical to reducing the risk of calculation errors.
• Generalization: Algebraic formulas, such as those used for arithmetic sequences, enable us to generalize and find not just specific terms but broader patterns in data.

Through algebra, you gain the ability to tackle both simple and complex summation problems effectively.