Problem 53 - Algebra and Trigonometry

Question

Write the sum without using sigma notation. $$\sum_{k=1}^{5} \sqrt{k}$$

Step-by-Step Solution

Answer

The sum is \( \sqrt{1} + \sqrt{2} + \sqrt{3} + \sqrt{4} + \sqrt{5} \).

Step 1: Understand the sigma notation

The expression \( \sum_{k=1}^{5} \sqrt{k} \) indicates that we need to find the sum of \( \sqrt{k} \) where \( k \) takes on each integer value from 1 to 5.

Step 2: Substitute values for k

First, substitute \( k = 1 \) into \( \sqrt{k} \), resulting in \( \sqrt{1} \). Continue this procedure for each value of \( k \) up to 5: \( \sqrt{2} \), \( \sqrt{3} \), \( \sqrt{4} \), and \( \sqrt{5} \).

Step 3: Write the series without sigma notation

The sum \( \sum_{k=1}^{5} \sqrt{k} \) translates to the expanded form \( \sqrt{1} + \sqrt{2} + \sqrt{3} + \sqrt{4} + \sqrt{5} \).

Step 4: Check and ensure all terms are included

Verify that all terms from \( \sqrt{1} \) to \( \sqrt{5} \) are included in the sum. Confirm that the series correctly represents the range of \( k \) values, ensuring no terms are missed or added.

Key Concepts

Understanding Series Expansion

The concept of series expansion often refers to the process of expressing a series in its detailed form. This makes it easier to understand and calculate. In mathematics, expanding a series means writing out all the terms individually. For example, if we have a series like \( \sum_{k=1}^{5} \sqrt{k} \), the series expansion would involve substituting integers for \( k \) one at a time.

This approach allows us to see each term separately and gain a better understanding of what is being summed up. In our case, this means writing it as \( \sqrt{1} + \sqrt{2} + \sqrt{3} + \sqrt{4} + \sqrt{5} \). Each term is calculated individually using their respective values of \( k \), before being added together to find the total sum of the series.

• A series expansion helps break down complex series expressions.
• It reveals all terms so calculations can be easily verified.
• This is crucial in cases where each term must be examined separately.

Understanding series expansion is important for clarity, especially in academic and real-world applications.

Basics of the Square Root

The square root is a fundamental mathematical operation that asks which number, when multiplied by itself, results in a given number. For example, the square root of 9 is 3, because \(3 \times 3 = 9\). Understanding the properties of square roots is crucial when working with series involving radicals, just like the one in the exercise.

Square roots appear in many mathematical problems and can often be simplified for easier computation. A radical, such as \( \sqrt{k} \), requires evaluating the square root of every integer value of \( k \) involved in the series expansion.

• The square root of a number \( x \) is denoted by \( \sqrt{x} \).
• It is only defined for non-negative numbers in the context of real numbers.
• Understanding square roots helps in simplifying complex mathematical expressions.

Grasping the basics of square roots is a key skill required in solving a variety of mathematical exercises. Interpretation becomes easier once you can securely handle square root operations.

The Concept of Sum of Series

The sum of a series is one of the central concepts in mathematics, especially when dealing with sequences or progressive calculations. This refers to the total obtained by adding all the terms in the series. In the original exercise, the sum of the series is expressed by the notation \( \sum_{k=1}^{5} \sqrt{k} \). Instead of calculations performed directly, sigma notation provides a compact form to express such sums.

When tackling such problems, breaking down the series into its terms helps achieve an accurate sum. Each term, such as \( \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5} \), is individually added to find the final result.

• The sum gives the combined total of all terms included in the expansion.
• Proper understanding is required to prevent error in adding up terms correctly.
• Using expanded form aids in visual verification of the sum.

Calculating the sum of series involves meticulous attention to each term, ensuring accurate results in both basic and complex scenarios.