Problem 120 - 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 expanded form is \( 4 + 9 + 16 \) and the total sum is 29.

Step 1: Understanding the Sigma Notation

The notation \( \sum_{k=3}^{5}(k-1)^{2} \) represents the sum of the expression \((k-1)^2\) evaluated for each integer \(k\) from 3 to 5. This means we need to substitute each value of \(k\) into the expression and then find the sum of those values.

Step 2: Evaluating the Expression for First Term

First, substitute \(k = 3\) into the expression \((k-1)^2\). Thus, \((3-1)^2 = 2^2 = 4\).

Step 3: Evaluating the Expression for Second Term

Next, substitute \(k = 4\) into the expression \((k-1)^2\). Thus, \((4-1)^2 = 3^2 = 9\).

Step 4: Evaluating the Expression for Third Term

Finally, substitute \(k = 5\) into the expression \((k-1)^2\). Thus, \((5-1)^2 = 4^2 = 16\).

Step 5: Writing the Sum in Expanded Form

Now, combine all the evaluated expressions to write the expanded form: \( 4 + 9 + 16 \). The sum of these terms is the result of the original sigma notation.

Step 6: Calculating the Total Sum

Add the numbers in the expanded form: \( 4 + 9 + 16 = 29 \). This is the total sum of the given expression.

Key Concepts

Expanded Form

When dealing with sigma notation, you might encounter situations where you need to find the expanded form of a series. But what exactly does expanded form mean? Simply put, expanded form is when you list out each individual term within a sequence or series instead of just the compact notation. This makes it easier to see and compute each step separately.

For instance, in the original exercise, we have the sigma notation \( \sum_{k=3}^{5}(k-1)^{2} \). The expanded form involves substituting each integer value from \(k = 3\) to \(k = 5\) into the expression \((k-1)^{2}\).

So, you calculate:

• When \(k = 3\), \((3-1)^2 = 4\).
• When \(k = 4\), \((4-1)^2 = 9\).
• When \(k = 5\), \((5-1)^2 = 16\).

Now, we've transformed the sigma notation into its expanded form: \(4 + 9 + 16\). This step allows us to clearly see each component that contributes to the final sum.

Sum of Series

Once you've converted a sigma notation into its expanded form, the next logical step is to find the sum of the series. A series is simply the sum of the terms of a sequence. In our example, the series consists of the numbers \(4, 9,\) and \(16\).

By adding these numbers together:

• \(4 + 9\) gives us \(13\).
• Adding \(16\) to \(13\) results in \(29\).

This result, \(29\), is the sum of the series from the original sigma notation.

The sum of the series is a crucial step as it provides the ultimate value when evaluating sigma notations. Whether dealing with arithmetic or geometric sequences, summing the expanded terms efficiently reveals the end result of the given problem.

Integer Substitution

In the context of sigma notation, integer substitution is a straightforward yet powerful technique that helps evaluate each term of a series. The process involves substituting each integer within a specified range into a given algebraic expression.

Take the original formula \((k-1)^2\) from our example. With integer substitution, you plug each whole number from 3 to 5 into this expression.

Steps include:

• Substituting \(k = 3\), you calculate \((3-1)^2 = 4\).
• For \(k = 4\), it becomes \((4-1)^2 = 9\).
• Lastly, \(k = 5\) results in \((5-1)^2 = 16\).

Each substitution replaces the variable with an actual number, allowing for an explicit numerical value instead of an abstract expression. This method simplifies understanding of how the sigma notation expands.

By systematically evaluating each term, integer substitution ensures comprehensive and clear representation of each part of the series, paving the way for accurate results in mathematical problems involving sums of sequences.