Problem 57 - Precalculus Mathematics for Calculus

Question

Write the sum without using sigma notation. $$\sum_{k=0}^{6} \sqrt{k+4}$$

Step-by-Step Solution

Answer

The sum without sigma notation is: 2 + \( \sqrt{5} \) + \( \sqrt{6} \) + \( \sqrt{7} \) + \( \sqrt{8} \) + 3 + \( \sqrt{10} \).

Step 1: Understand the Problem

The expression involves a summation from 0 to 6 of the term \( \sqrt{k+4} \). The goal is to express this summation without using sigma notation.

Step 2: Identify the Terms of the Summation

Sigma notation expresses a summation of several terms. Here, the terms are \( \sqrt{k+4} \) for \( k \) ranging from 0 to 6. Calculate each term separately by substituting each value of \( k \) in the expression.

Step 3: Calculate Each Term

Let's calculate:- For \( k = 0 \), the term is \( \sqrt{0+4} = \sqrt{4} = 2 \).- For \( k = 1 \), the term is \( \sqrt{1+4} = \sqrt{5} \).- For \( k = 2 \), the term is \( \sqrt{2+4} = \sqrt{6} \).- For \( k = 3 \), the term is \( \sqrt{3+4} = \sqrt{7} \).- For \( k = 4 \), the term is \( \sqrt{4+4} = \sqrt{8} \).- For \( k = 5 \), the term is \( \sqrt{5+4} = \sqrt{9} = 3 \).- For \( k = 6 \), the term is \( \sqrt{6+4} = \sqrt{10} \).

Step 4: Write the Sum Without Sigma Notation

Now, write the entire sum as an explicit expression without the sigma notation. The sum is:\[2 + \sqrt{5} + \sqrt{6} + \sqrt{7} + \sqrt{8} + 3 + \sqrt{10}\]

Step 5: Verify the Calculation

Verify each term calculated from the range of \( k \), ensuring no step skipped or mistaken. All calculations should make the process straightforward to match with the sigma notation starting expression.

Key Concepts

Summation

Summation is a mathematical operation that adds up a sequence of numbers. It is represented by the Greek letter sigma (\( \Sigma \)), hence the term "sigma notation." When you use sigma notation, you have a clearer and more compact way to express the addition of multiple terms. The notation includes:

• A starting index, which is the beginning number of the sequence.
• An ending index, which shows where the sequence stops.
• A general term for the sequence, which might include variables such as \( k \).

In the example \( \sum_{k=0}^{6} \sqrt{k+4} \), the summation is from \( k = 0 \) to \( k = 6 \), and the term is \( \sqrt{k+4} \) for each \( k \). This means we calculate each term by substituting different values of \( k \), and then add them together.

Square Root

The square root is a mathematical function that finds a number which, when multiplied by itself, equals the given number. It is denoted by the symbol \( \sqrt{} \). Calculating the square root is straightforward:

• The square root of 4, \( \sqrt{4} \), is 2 because \( 2 \times 2 = 4 \).
• Square roots of numbers not perfect squares (e.g., \( \sqrt{5} \)), result in irrational numbers that cannot be expressed as a simple fraction.

Understanding the square root is essential in solving various mathematical problems, as it frequently appears in formulas and equations, especially when dealing with areas and volumes.

Mathematics Problem Solving

Mathematics problem solving requires understanding the problem, formulating a plan, carrying out that plan, and reviewing the solution. Let’s break this process into steps:

• Understanding the Problem: Identify what is being asked and the information available.
• Formulating a Plan: Determine the steps or operations needed to find the solution.
• Carrying Out the Plan: Execute the steps logically, ensuring each step follows from the previous.
• Reviewing: Check if the solution makes sense and meets the original requirements.

For the exercise provided, the goal was to express the summation without using sigma notation. By calculating each term in the sequence and then summing them up, we successfully solved the problem.

Step by Step Solution

A step by step solution helps in guiding through complex problems by breaking them into manageable pieces. These steps should be clear and logical to aid understanding:

• Step 1: Grasp the problem. Here, understand that the summation needs to be expressed in a different form.
• Step 2: Recognize the terms. Identify the series of operations or sequence involved, such as \( \sqrt{k+4} \) for different \( k \).
• Step 3: Calculate each step carefully. Solve each term one by one, ensuring accuracy (e.g., \( \sqrt{4} = 2 \), \( \sqrt{9} = 3 \)).
• Step 4: Combine all results without sigma notation. Write out the expression explicitly (e.g., \[2 + \sqrt{5} + \sqrt{6} + \sqrt{7} + \sqrt{8} + 3 + \sqrt{10}\]).

This methodical approach not only helps in solving this particular problem but also aids in tackling similar problems in mathematics.