Problem 53 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Sigma Notation

The sigma notation \(\sum_{k=1}^{5} \sqrt{k}\) represents a summation where \(k\) starts at 1 and increases by 1 each time until it reaches 5. For each value of \(k\), you calculate \(\sqrt{k}\) and then add those results together.

Step 2: Calculate Each Term Individually

Calculate the value of \(\sqrt{k}\) for each integer from 1 to 5:- \(\sqrt{1} = 1\)- \(\sqrt{2} = \sqrt{2}\)- \(\sqrt{3} = \sqrt{3}\)- \(\sqrt{4} = 2\)- \(\sqrt{5} = \sqrt{5}\)

Step 3: Write the Sum Without Sigma Notation

Write the expression as a sum of all individual terms calculated in the previous step:\[1 + \sqrt{2} + \sqrt{3} + 2 + \sqrt{5}\]

Key Concepts

Understanding Summation

Summation is a mathematical operation that refers to the addition of a sequence of numbers. When we use summation, we often aggregate values following a specific rule or formula. In the context of sigma notation, summation is a compact way of expressing the addition of terms generated by a specific function or formula. For example:

• The sigma symbol \(\sum\) indicates the start of a sum.
• An index of summation (like \(k\) in our problem) defines the variable being summed over.
• The limits of summation, such as \(k=1\) to \(k=5\), state the starting and ending values.

In practical scenarios, summation helps in breaking down complex expressions into simpler components. This makes them easier to handle computationally and conceptually. It also allows mathematicians and students to understand patterns within sequences and series, making this fundamental concept vital in mathematics.

Exploring Square Roots

Square roots are a basic yet crucial concept in mathematics that represent a value which, when multiplied by itself, gives the original number. For instance, the square root of 4 is 2 because \(2 \times 2 = 4\). In the context of our specific problem, we're taking square roots of integers from 1 to 5.Here are key points about square roots:

• Notation: The square root of a number \(k\) is denoted as \(\sqrt{k}\).
• Properties: Square roots offer insights into quadratic equations and geometry.
• Calculation: Simple numbers such as 1, 4, 9, 16, etc., have straightforward square roots (e.g., \(\sqrt{4} = 2\)), while others, like \(\sqrt{5}\), are irrational and need approximation.

Understanding square roots extends beyond simple computation. It enriches a grasp of correlation in geometry and algebra, allowing deeper dives into mathematical theory and practices.

Unpacking Notation

Notation in math is a system of symbols used to represent numbers, operations, and concepts concisely. The power of notation lies in its ability to simplify complex mathematical ideas into comprehensible terms. For sigma notation, several key elements require understanding:

• Sigma (\(\sum\)) signifies the operation of summation and is a short, elegant way to depict the addition of sequences.
• The variable or index (like \(k\)) acts as a placeholder that changes its value according to the specified limits in the notation.
• The expression following the sigma symbol describes the operation performed for each value of the variable (in this case, \(\sqrt{k}\)).

By mastering the notation, mathematicians and students can easily manage and manipulate complex expressions in concise forms, whether in sequences, series, or other mathematical structures. This understanding is fundamental in advancing one's mathematical knowledge and solving higher-level problems with efficiency and clarity.