Problem 6 - Calculus for Biology and Medicine

Question

Write each sum in expanded form. $$ \sum_{k=3}^{5}(k-1)^{2} $$

Step-by-Step Solution

Answer

The sum in expanded form is \(4 + 9 + 16\).

Step 1: Understand the Summation Notation

The expression \(\sum_{k=3}^{5}(k-1)^{2}\) means that you should evaluate \((k-1)^{2}\) for each integer \(k\) from 3 to 5 and then sum the results. So we need to plug in \(k = 3, 4,\) and \(5\) into \((k-1)^{2}\).

Step 2: Calculate for k = 3

Substitute \(k = 3\) into the expression \((k-1)^{2}\):\[(3-1)^{2} = 2^{2} = 4\].

Step 3: Calculate for k = 4

Substitute \(k = 4\) into the expression \((k-1)^{2}\):\[(4-1)^{2} = 3^{2} = 9\].

Step 4: Calculate for k = 5

Substitute \(k = 5\) into the expression \((k-1)^{2}\):\[(5-1)^{2} = 4^{2} = 16\].

Step 5: Write the Sum in Expanded Form

Now that we have calculated each term, the expanded form of the sum is:\[4 + 9 + 16\].

Key Concepts

Expanded Form

In mathematics, when we talk about writing an expression in expanded form, we're essentially breaking down and expressing longer operations in their individual components. Let's consider the summation expression \( \sum_{k=3}^{5}(k-1)^{2} \). This uses the Greek letter Sigma to indicate that we're going to sum several terms. To put it in expanded form, you evaluate the expression for every integer within the specified range, which in this case is from 3 to 5.

• Start by substituting \( k = 3 \) into \( (k-1)^2 \), which gives us 4.
• Then substitute \( k = 4 \), resulting in 9.
• Finally, substitute \( k = 5 \), resulting in 16.

So the expanded form of \( \sum_{k=3}^{5}(k-1)^{2} \) is \( 4 + 9 + 16 \). This representation helps in understanding each individual term's contribution to the total sum.

Mathematical Series

A mathematical series is created when you add the terms of a sequence. With summation notation, we're able to represent a series compactly and efficiently. In our example, \( \sum_{k=3}^{5}(k-1)^{2} \), we're summing the squares of integers adjusted by 1. A process like this involves:

• Identifying each term through the given expression \((k-1)^2\) as our series' pattern or rule.
• Summing the calculated terms: \(4 + 9 + 16\), which finally gives us the total sum of 29.

Series help in understanding the aggregate result of sequences of operations. They also play an essential role in higher mathematics, aiding calculations in calculus and other advanced fields.

Integer Evaluation

Evaluation of integers in given expressions is a fundamental skill in mathematics. It involves substituting whole numbers into expressions to calculate their values. This process is vital in achieving accurate results when working with summation notation. For example, in our expression \( \sum_{k=3}^{5}(k-1)^{2} \), integer evaluation comes when:

• \( k = 3 \) gives the result of \( (3-1)^2 = 4 \).
• \( k = 4 \) gives the result of \( (4-1)^2 = 9 \).
• \( k = 5 \) gives the result of \( (5-1)^2 = 16 \).

This step-by-step evaluation ensures each part of the mathematical calculation is correctly understood and contributes accurately to the final sum. Integer evaluation is particularly important in educational settings for building foundational understanding in mathematics.