Problem 34 - Algebra and Trigonometry

Question

find each indicated sum. $$ \sum_{k=1}^{4}(k-3)(k+2) $$

Step-by-Step Solution

Answer

The sum is -4.

Step 1: Evaluate the function at each value of k

Start by substituting k = 1, k = 2, k = 3, and k = 4 into the expression \( (k-3)(k+2) \): \nFor k = 1, \( (k-3)(k+2) = (-2)(3) = -6 \) \nFor k = 2, \( (k-3)(k+2) = (-1)(4) = -4 \) \nFor k = 3, \( (k-3)(k+2) = (0)(5) = 0 \) \nFor k = 4, \( (k-3)(k+2) = (1)(6) = 6 \)

Step 2: Sum up all the results

The summation is the total sum of these evaluated expressions. So add all the results together: \n\(-6 - 4 + 0 + 6 = -4 \)

Key Concepts

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operations. These expressions form the basis for understanding how to manipulate equations and solve problems. In the given exercise, the expression \((k-3)(k+2)\) is an example of a simple polynomial which involves the variable \(k\).

Algebraic expressions often require evaluation by substituting different values for variables. This helps in determining numerical outcomes, as seen in our exercise where different values of \(k\) are plugged into the expression.

• Constants like \(-3\) and \(+2\) do not change and are used to scale or shift the values of \(k\).
• Each value of \(k\) changes the overall result of the expression, affecting the sum.

Understanding how these operations affect expressions enhances problem-solving skills in various math contexts.

Sigma Notation

Sigma notation is a compact way to write the sum of a sequence of terms. This mathematical symbol \(\Sigma\) signifies summation, helping to easily convey the addition of many terms without writing them all out.

In the exercise, you see \(\sum_{k=1}^{4}(k-3)(k+2)\). This tells us to evaluate the expression \((k-3)(k+2)\) for each integer value of \(k\) from 1 to 4.

• The symbol \(\Sigma\) dictates that we need to sum all the evaluated terms.
• The limits of summation (1 to 4) indicate which values of \(k\) are used.
• Each term is calculated individually, and then all terms are added together to find the total sum.

This method simplifies complex additions, making calculations straightforward and structured.

Sequences and Series

The concepts of sequences and series are closely linked, and they help in understanding patterns and summations. A sequence is a list of numbers in a specific order, and a series is the sum of a sequence's terms.

In our problem, the expression is evaluated for a sequence of values: \(k = 1, 2, 3, 4\). Each calculated value \(-6, -4, 0, 6\) forms part of this sequence.

• A sequence can be finite, like in this exercise with four terms, or infinite.
• A series then sums these terms to produce a total, which is \(-4\) in this case.
• Knowing the regularity in sequences helps in predicting patterns and solving summation problems efficiently.

Understanding these concepts allows for solving both simple and complex mathematical problems, developing a deeper grasp of patterns and mathematical operations.