Problem 55 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\( \sqrt{4} + \sqrt{5} + \sqrt{6} + \sqrt{7} + \sqrt{8} + \sqrt{9} + \sqrt{10} \)

Step 1: Understand the Sigma Notation

The given sigma notation \( \sum_{k=0}^{6} \sqrt{k+4} \) represents the sum where the variable \( k \) starts at 0 and ends at 6. For each integer value of \( k \), substitute it into the expression \( \sqrt{k+4} \) to find the individual terms.

Step 2: Substitute Values and List Terms

Starting with \( k = 0 \), substitute into \( \sqrt{k+4} \) to get \( \sqrt{4} \). Continue substituting values up to \( k = 6 \). This gives the sequence of terms: \( \sqrt{4}, \sqrt{5}, \sqrt{6}, \sqrt{7}, \sqrt{8}, \sqrt{9}, \sqrt{10} \).

Step 3: Write the Sum Without Sigma Notation

Combine the terms obtained from Step 2 into a single expression without the use of sigma notation: \( \sqrt{4} + \sqrt{5} + \sqrt{6} + \sqrt{7} + \sqrt{8} + \sqrt{9} + \sqrt{10} \).

Key Concepts

Summation

Summation is a concept in mathematics that involves adding a sequence of numbers. It is represented using the Greek letter sigma (\( \sum \)). This operation allows you to sum terms that follow a particular rule or pattern. In the context of the problem, summation is used to add together all the square roots of numbers formed by the expression \( \sqrt{k+4} \) for each integer \( k \) from 0 to 6.

Each integer value of \( k \) gives rise to a unique term that contributes to the overall sum. The role of summation here is to simplify the process of adding multiple terms by having a concise mathematical notation that describes all these operations. It provides a compact and efficient way to present large and complex sums.

Mathematical Series

A mathematical series is the sum of the terms of a sequence. It is crucial to distinguish between a sequence and a series. A sequence is a list of numbers in a particular order, while a series is the sum of these numbers.

In a typical mathematical series, such as the one in our exercise, the terms are the individual square root values calculated for each integer \( k \). These terms form a progressive sequence. Once these are combined using addition, we achieve a series. Understanding the relationship between sequences and series is fundamental in analyzing and solving problems involving such progressions.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. In our exercise, the function \( \sqrt{k+4} \) forms the basis of our sequence. This function increases sequentially as \( k \) increases from 0 to 6.

The concept of square roots is essential in simplifying expressions and equations. Here, calculating \( \sqrt{4}, \sqrt{5}, \sqrt{6}, \sqrt{7}, \sqrt{8}, \sqrt{9}, \) and \( \sqrt{10} \) requires recognizing that these represent the potential dimensions or roots of perfect squares or nearer values. Understanding how to compute square roots, especially with non-perfect squares, enhances your algebraic and arithmetic skills.

Expression Simplification

Expression simplification involves using rules of algebra to rewrite an expression in a simpler or more standard form. The goal is to make the expression easier to understand or work with.

In the context of this exercise, simplifying the expression means writing out the individual terms formed by \( \sqrt{k+4} \) for each value of \( k \). By doing so, we remove the sigma notation and represent the sum directly as \( \sqrt{4} + \sqrt{5} + \sqrt{6} + \sqrt{7} + \sqrt{8} + \sqrt{9} + \sqrt{10} \).

This process not only eliminates the compact yet abstract form of sigma notation but also gives a clearer, step-by-step insight into how each term contributes to the total sum. Mastery of this technique is particularly useful in simplifying more complex mathematical expressions.