Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

\(\sum_{k=0}^{4}(k-1)(k-3)\)

Step-by-Step Solution

Answer

The sum is 5.

Step 1: Understanding the Problem

We need to evaluate the summation: \(\sum_{k=0}^{4}(k-1)(k-3)\). This means adding the values of the expression \((k-1)(k-3)\) for all integer values of \(k\) from \(0\) to \(4\).

Step 2: Expand the Expression

For each \(k\), we expand \((k-1)(k-3)\). The expression can be expanded as follows: \((k-1)(k-3) = k^2 - 3k - 1k + 3 = k^2 - 4k + 3\).

Step 3: Calculate Each Term

Next, we calculate each term from \(k = 0\) to \(4\):- For \(k=0\): \((0-1)(0-3) = 1 \cdot 3 = 3\)- For \(k=1\): \((1-1)(1-3) = 0\)- For \(k=2\): \((2-1)(2-3) = 1 \cdot (-1) = -1\)- For \(k=3\): \((3-1)(3-3) = 0\)- For \(k=4\): \((4-1)(4-3) = 3 \cdot 1 = 3\)

Step 4: Sum the Calculated Terms

Now, we sum the values obtained from each term:\[3 + 0 - 1 + 0 + 3 = 5\].

Key Concepts

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial as a sum of terms with each term being a product of a coefficient and a variable raised to an exponent. In the context of our exercise, we expanded the expression

• \((k-1)(k-3)\)

by multiplying each term in one polynomial by each term in the other polynomial.This method allows us to simplify complex expressions by using basic arithmetic operations: addition and multiplication. For instance, in our problem,

• expanding \((k-1)(k-3)\)allowed us to express it as \(k^2 - 4k + 3\).

Understanding polynomial expansion helps in many algebraic procedures, such as integrating polynomial expressions or finding roots.

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant. In the original exercise, we're summing different terms obtained by plugging values from \(k = 0\) to \(k = 4\) into the expanded polynomial.Although this isn't a typical arithmetic sequence, understanding arithmetic sequences can make it easier to deal with series. This is because many properties used in arithmetic sequences also apply to other sequences and series in algebra. For example, by observing the changes in the polynomial values over a range, we can better understand how the sequence behaves.Using these basic principles is crucial in solving more complicated algebra problems.

Mathematical Reasoning

Mathematical reasoning is the ability to use mathematical principles and logic to deduce solutions or prove statements. In the context of our problem, we use mathematical reasoning to understand the steps needed to evaluate the summation. First, we use logical reasoning to expand the polynomial expression. Next, we calculate each term individually by substituting values from zero to four into the expression. Then, we sum all evaluated terms to finalize the answer, which requires checking that each calculation is logically consistent. Additionally, mathematical reasoning helps ensure that all steps adhere to mathematical rules and principles. This skill is essential for solving equations, proving theorems, and performing mathematical operations correctly.