91Ó°ÊÓ

Let's dive into the concept of evaluating expressions. In the given problem, we have an expression to evaluate: \(k^2 - 2k + 3\).
To start, you need to substitute each value of \(k\) from the specified range (0 to 5) into the expression.
This step-by-step approach helps ensure that we do not miss any values. For instance, when \(k = 0\), we consider \(0^2 - 2(0) + 3 = 3\).
Similarly, when \(k = 1\), it becomes \(1^2 - 2(1) + 3 = 2\). This process is vital because it lets us understand how the function behaves across different inputs and how each term contributes to the final sum.

A series is essentially the sum of terms that follow a specific rule or pattern. In our exercise, the terms of the series are generated by the expression \(k^2 - 2k + 3\) for \(k\) ranging from 0 to 5.
To find the series, we first generate the sequence by calculating each term for the values of \(k \in [0, 1, 2, 3, 4, 5]\).
These terms are \3, 2, 3, 6, 11,\ and \18\ respectively.
The series is then the sum of these terms, which gives us the final value of 43.
Understanding series is crucial because it lays the groundwork for deeper concepts in calculus and advanced mathematics. It is also frequently used in various real-world applications, including economics and physics.

Simplification helps to reduce expressions or functions to a more manageable form. In the context of our problem, simplification mainly involves the easy evaluation and eventual summation of each term.
Once each term \(k^2 - 2k + 3\) is independently evaluated for each integer \(k\) from 0 to 5, it simplifies our task. Each calculated term \(3, 2, 3, 6, 11,\ and \18\) is added together for the final sum.
Good simplification practices ensure that complex problems become easier to handle, step by step.
Rather than getting overwhelmed, taking a structured approach as shown in our steps makes it clearer and simpler.
This principle is widely applicable in various problem-solving scenarios in mathematics and beyond.