Problem 127 - Precalculus with Limits

Question

In Exercises 127 and 128, determine whether the statement is true or false. Justify your answer. \( \displaystyle \sum_{i=1}^{4} (i^2 + 2i) = \sum_{i=1}^{4} i^2 + 2 \sum_{i=1}^{4} i \)

Step-by-Step Solution

Answer

The statement is true because both sides of the equation are equal to 50.

Step 1: Compute the left side of the equation

The left hand side (LHS) of the equation involves computing the sum of \(i^2 + 2i\) for \(i = 1\) to \(4\). This will give: \(\sum_{i=1}^{4} (i^2 + 2i) = 1^2 + 2*1 + 2^2 + 2*2 + 3^2 + 2*3 + 4^2 + 2*4 = 1 + 2 + 4 + 4 + 9 + 6 + 16 + 8 = 50\)

Step 2: Compute the right side of the equation

The right hand side (RHS) of the equation involves computing two separate sums: \( \sum_{i=1}^{4} i^2 \) and \(2\sum_{i=1}^{4} i\). Calculating these individually, we get: \( \sum_{i=1}^{4} i^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30 \) and \( 2\sum_{i=1}^{4} i = 2(1 + 2 + 3 + 4) = 2(10) = 20 \). So the RHS of the equation is \( 30 + 20 = 50 \)

Step 3: Compare the results

After computing both sides of the equation, compare the results. It is seen that both the LHS and the RHS are equal to 50. Therefore, the given statement is true.

Key Concepts

Sigma Notation

Sigma notation, denoted by the Greek letter sigma \( \Sigma \), streamlines the process of adding up a long series of numbers. It provides a concise way to represent the sum of a sequence of terms defined by a formula. For example, to add up the first four square numbers plus twice the sequence numbers, you would write \( \sum_{i=1}^{4} (i^2 + 2i) \). Here, the \(i \) underneath the \Sigma indicates the starting index, while the number 4 at the top is the ending index; together they define the range of summation. The expression \(i^2 + 2i \) to the right of the \Sigma shows the general term to be added.

Breaking down sigma notation into individual components helps better understand and solve the series. In the exercise, by separately evaluating each term and summing them, we confirmed the property that \( \sum_{i=1}^{4} (i^2 + 2i) = \sum_{i=1}^{4} i^2 + 2\sum_{i=1}^{4} i \). This equation shows how a single sigma notation can be separated into the sum of two distinct sigma notations under certain conditions.

Series and Sequences

A sequence is an ordered list of numbers following a specific rule, while a series is the sum of the elements of a sequence. In mathematics, distinguishing between these two is crucial. For instance, we have a sequence of square numbers \(1, 4, 9, 16\) and a series when the same numbers are added together \(1+4+9+16\).

Understanding the relationship between individual terms of a sequence and their sum is often a staple in mathematical analysis. In the solved exercise, the sequence is generated by the term \(i^2 + 2i\), while the series is the sum indicated by the sigma notation. The careful observation of how sequences unfold into series using sigma notation can help students apply the appropriate properties of summation to simplify and solve equations involving series.

Proofs in Mathematics

Proofs form the backbone of mathematical certainty. They confirm the truth or falsehood of statements by logical reasoning from known facts and accepted principles. The solution provided for the exercise serves as a direct proof, where we started with the given equation and performed a step-by-step verification to confirm its truth.

To make the exercise more comprehensible and user-friendly, realising that there are different types of proofs can be helpful—such as direct proof, inductive proof, contradiction, and counterexample. In this scenario, we used a direct proof where both sides of the equation were separately calculated and then compared to demonstrate the validity of the initial statement. Mastering different proof techniques can greatly enhance students' problem-solving skills in mathematics.