Problem 67 - College Algebra with Modeling and Visualization

Question

Write out the terms of the series and then evaluate it. $$\sum_{k=4}^{5}\left(k^{2}-k\right)$$

Step-by-Step Solution

Answer

The sum of the series is 32.

Step 1: Interpret the Summation

The summation \( \sum_{k=4}^{5} (k^2 - k) \) means we will calculate the expression \( k^2 - k \) for each integer value of \( k \) from 4 to 5, and then add those results together.

Step 2: Calculate the First Term (k=4)

Substitute \( k = 4 \) into the expression \( k^2 - k \). Calculate \( 4^2 - 4 = 16 - 4 = 12 \). So, the first term is 12.

Step 3: Calculate the Second Term (k=5)

Substitute \( k = 5 \) into the expression \( k^2 - k \). Calculate \( 5^2 - 5 = 25 - 5 = 20 \). So, the second term is 20.

Step 4: Sum the Terms

Add the terms that we calculated: 12 + 20. This results in 32.

Step 5: Final Evaluation

The evaluation of the series \( \sum_{k=4}^{5} (k^2 - k) \) is complete. The sum of the terms 12 and 20 gives us the final answer, which is 32.

Key Concepts

Summation Notation

Summation notation is a concise way to express the sum of a sequence of numbers. Instead of writing out each term and then adding them together, this notation lets us use a symbol to signify the addition of many terms. The Greek letter sigma (\( \Sigma \)) is used to represent this operation.
Think of it as a shorthand for adding several numbers together.

• The expression under the sigma tells us what to do with the variable.
• The upper and lower bounds of the sigma tell us which numbers to use for the variable.

In the example \( \sum_{k=4}^{5} (k^2 - k) \), the lower bound is 4, and the upper bound is 5. This means we substitute the values 4 and 5 for \( k \) into our expression\( k^2 - k \).

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations.
There are no equals signs in an algebraic expression, so they are more like phrases.
In \( k^2 - k \), \( k \) is our variable.

• When \( k \) takes on a value, like 4 or 5, the expression can yield a specific number.
• Operations like squaring and subtracting can be seen here, which change the numbers.

In this series evaluation example, the algebraic expression is applied to each integer in the range to create the terms that are summed.

Mathematical Series

A mathematical series is simply the sum of the terms of a sequence. This falls under a wider category known as sequences and series, which is a common topic in algebra and calculus. The series we evaluated,\( \sum_{k=4}^{5} (k^2 - k) \),is the sum of our calculated terms for each integer between the lower and upper bounds (here, 4 and 5).

• The terms are derived by substituting the integer values into the algebraic expression.
• Each evaluated expression contributes to the overall sum of the series.

In problems like these, a series can be finite (with a specific number of terms) or infinite. Our example was finite because we only looked at two integers.

Integer Substitution

Substitution is a mathematical process of replacing a variable with a number. It is key in evaluating algebraic expressions and solving equations. To substitute in the context of our series:

• First, take each value of\( k \)in the given range (here, 4 and 5).
• Insert that value into your expression,\( k^2 - k \).
• Calculate to find the result for each specific integer.

Without substitution, we would not be able to evaluate each term of the expression and thus could not find the sum of the series. In our exercise, we substituted 4 and 5 into the expression to find the values 12 and 20, respectively, showing the power of this simple yet effective technique.