Problem 5 - Calculus for Biology and Medicine

Question

Write each sum in expanded form. $$ \sum_{k=1}^{4} \sqrt{k} $$

Step-by-Step Solution

Answer

The summation expands to \( 1 + 1.414 + 1.732 + 2 \).

Step 1: Understanding the Summation Notation

The given expression \( \sum_{k=1}^{4} \sqrt{k} \) is a summation notation. It tells us to add up expressions for each integer value of \( k \) starting from 1 and ending at 4.

Step 2: Expanding the Summation

To expand \( \sum_{k=1}^{4} \sqrt{k} \), substitute each integer value from 1 to 4 into the expression \( \sqrt{k} \). This gives us the sequence: \( \sqrt{1} + \sqrt{2} + \sqrt{3} + \sqrt{4} \).

Step 3: Calculating Each Term

Calculate each term separately: - \( \sqrt{1} = 1 \) - \( \sqrt{2} \) is approximately 1.414 - \( \sqrt{3} \) is approximately 1.732 - \( \sqrt{4} = 2 \)

Step 4: Assembling the Expanded Form

The expanded form of the summation is approximated as follows: \[ 1 + 1.414 + 1.732 + 2 \]

Key Concepts

Expanded Form

Expanded form is a method of expressing a sum or expression by writing out each term separately rather than using a shorter notation. In the context of summation notation, it allows us to visualize each component of the expression, which can be helpful for understanding and calculation.
When dealing with summation, such as in the problem \( \sum_{k=1}^{4} \sqrt{k} \), you replace the variable \( k \) with each integer from the starting point to the endpoint. This results in listing all the terms needed to satisfy the original expression.
This is done as follows:

• Replace \( k \) with 1: \( \sqrt{1} \)
• Replace \( k \) with 2: \( \sqrt{2} \)
• Replace \( k \) with 3: \( \sqrt{3} \)
• Replace \( k \) with 4: \( \sqrt{4} \)

So, the expanded form of the above expression is \( \sqrt{1} + \sqrt{2} + \sqrt{3} + \sqrt{4} \), making each step of the summation clear and understandable.

Square Root

The square root is a number, which when multiplied by itself, gives the original number. The square root of a number \( x \) is often represented by \( \sqrt{x} \).
Understanding square roots is essential when dealing with problems involving summation with square roots, as calculating each term separately can make the summation process easier.
For integer values:

• \( \sqrt{1} \) = 1, because \(1 \times 1 = 1\)
• \( \sqrt{4} \) = 2, because \(2 \times 2 = 4\)

However, not all numbers have an integer square root. For example,

• \( \sqrt{2} \approx 1.414 \),
• \( \sqrt{3} \approx 1.732 \)

These approximations help us work with irrational numbers, which are numbers that cannot be expressed as simple fractions.

Sequence

A sequence is an ordered list of numbers following a specific rule or pattern. In mathematics, sequences are a way of showing elements arranged in a specific order.
When dealing with summation, a sequence is formed by evaluating each term separately. Our given problem, \( \sum_{k=1}^{4} \sqrt{k} \), creates a sequence of square roots:

• First term: \( \sqrt{1} \)
• Second term: \( \sqrt{2} \)
• Third term: \( \sqrt{3} \)
• Fourth term: \( \sqrt{4} \)

This set of numbers, arranged in a particular pattern, is a simple example of a mathematical sequence. Each term in the sequence corresponds to a different value of \( k \) in the summation. Identifying the sequence helps in organizing and structuring information during the evaluation process.

Integer Values

Integer values are whole numbers that can be either positive, negative, or zero, without fractional or decimal parts. When working on summation problems, it is common to encounter integer values, as they often define the limits and increments for the sequence.
For the sum \( \sum_{k=1}^{4} \sqrt{k} \), integer values define the range of \( k \) from 1 to 4, including all values in between. This range indicates which integers will substitute for \( k \) in the square root expression.
Each integer in the sequence represents a distinct point to evaluate the summation:

• 1: Leading to \( \sqrt{1} \),
• 2: Leading to \( \sqrt{2} \),
• 3: Leading to \( \sqrt{3} \),
• 4: Leading to \( \sqrt{4} \)

This orderly progression emphasizes the role integer values play in forming the framework of a structured mathematical sequence or summation.