Problem 57 - Algebra and Trigonometry

Question

\(55-60\) . Write the sum without using sigma notation. $$ \sum_{k=0}^{6} \sqrt{k+4} $$

Step-by-Step Solution

Answer

The sum is \( \sqrt{4} + \sqrt{5} + \sqrt{6} + \sqrt{7} + \sqrt{8} + \sqrt{9} + \sqrt{10} \).

Step 1: Understanding the Problem

We are asked to write the sum represented in sigma notation \( \sum_{k=0}^{6} \sqrt{k+4} \) as an expanded expression without sigma notation. This means we'll need to evaluate the expression inside the square root for each integer value of \( k \) ranging from 0 to 6, and then write out the terms separately.

Step 2: Expanding the Terms

Substitute each integer value from 0 to 6 into the expression \( \sqrt{k+4} \). This will result in separate terms for each evaluation:- For \( k = 0 \), the term is \( \sqrt{0+4} = \sqrt{4} \).- 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 \( \sqrt{5+4} = \sqrt{9} \).- For \( k = 6 \), the term is \( \sqrt{6+4} = \sqrt{10} \).

Step 3: Writing the Expanded Expression

Now that we have evaluated each \( \sqrt{k+4} \) for \( k \) from 0 to 6, we can write the expanded expression:\( \sqrt{4} + \sqrt{5} + \sqrt{6} + \sqrt{7} + \sqrt{8} + \sqrt{9} + \sqrt{10} \).

Key Concepts

Expanded Form

Expanded form is all about expressing a complex mathematical notation in a simpler and more detailed way. When dealing with sigma notation, like the one in the exercise, our goal is to unpack it so that all individual terms in a series are clearly displayed.

Sigma notation is often used to express sums in a concise form, but it can be intimidating or confusing if you're not familiar with it. In our exercise, the aim is to convert the sum represented by the sigma notation \( \sum_{k=0}^{6} \sqrt{k+4} \) into an expanded form. This involves evaluating the expression inside the sigma for each value in the range of the index.

To expand:

• Start from the lower limit of the sum (\(k = 0\)) and go up to the upper limit (\(k = 6\)).
• For each \(k\), replace \(k\) in the expression \(\sqrt{k+4}\) and calculate the term.
• List each term separately to form the expanded expression: \( \sqrt{4} + \sqrt{5} + \sqrt{6} + \sqrt{7} + \sqrt{8} + \sqrt{9} + \sqrt{10} \).

This approach gives you a clear view of how the sum unfolds, and helps demystify the sigma notation by revealing every part of the series.

Sum of Series

The sum of a series is simply the total of all its individual terms. When we work with sigma notation, our objective is to find this sum, but first, we need to know what we are adding up!

In our exercise, after converting the sigma notation to its expanded form, we have a series: \( \sqrt{4} + \sqrt{5} + \sqrt{6} + \sqrt{7} + \sqrt{8} + \sqrt{9} + \sqrt{10} \). Summing these terms provides a deeper understanding, because now we've spelled out each piece of the total.

The process involves:

• Listing all terms, which we've already done.
• Adding them together meticulously.

Remember, finding the sum may not always require direct addition, especially when terms can be simplified or remain in a radical form as many do in this case. Understanding each component of the sum and adding them precisely is key to mastering series.

Square Root Evaluation

Square roots can often seem tricky at first glance, as they involve finding a number which, when multiplied by itself, gives the original number under the square root symbol \( \sqrt{} \).

In our problem, each term involves evaluating a square root, which is part of understanding the sum of the series. For some values, like \( \sqrt{4} \) and \( \sqrt{9} \), the evaluation is straightforward because they are perfect squares, resulting in 2 and 3 respectively.

Here’s how to evaluate square roots as in our expanded list:

• Identify perfect squares, which simplify neatly (\( \sqrt{4} = 2 \) and \( \sqrt{9} = 3 \)).
• For non-perfect squares, like \( \sqrt{5}, \sqrt{6}, \sqrt{7}, \sqrt{8}, \sqrt{10} \), the evaluation generally results in an irrational number that doesn’t simplify to a neat integer.

By carefully evaluating each square root, you can better appreciate the structure of series and engage with their sometimes complex arithmetic properties. Plus, this practice enriches your understanding of irrational numbers and offers a valuable challenge in mathematical thinking.