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In algebra, substitution is a method used to solve equations or transform expressions by replacing variables with other expressions or values. It’s like swapping parts of a puzzle to see how it changes the outcome. When you perform substitution, you often replace a variable with another variable or a number that represents it. In the original exercise, we replaced every instance of the variable \(k\) with \(k+1\) in the function \(P_k = \frac{5}{k(k+1)}\).

Here's a simple process to follow when doing substitution in algebra:

• Identify the variable you want to replace.
• Determine what you will substitute in its place.
• Perform the substitution carefully, ensuring every instance of the variable is replaced.

The goal is often to find a new or altered equation, as we did with \(P_{k+1} = \frac{5}{(k+1)((k+1) + 1)}\), which provided us a new expression for evaluating the sequence.

Simplification is the process of making a mathematical expression easier to understand or use by breaking it down into simpler parts. In mathematics, this is key in solving and presenting problems more efficiently and clearly.

In the exercise, to simplify the term \(P_{k+1} = \frac{5}{(k+1)((k+1) + 1)}\), the expression \((k+1) + 1\) in the denominator was simplified to \(k+2\). The simplified form of the function becomes \(P_{k+1} = \frac{5}{(k+1)(k+2)}\).

Here are some tips for simplification:

• Follow the order of operations: parentheses, exponents, multiplication and division, addition and subtraction (PEMDAS).
• Look for opportunities to combine like terms or reduce fractions.
• Check your work, verifying that the simplified expression is equivalent to the original.

Simplification is a crucial skill for making calculations more practical and expressions more concise.

Sequences and series are fundamental concepts in mathematics that deal with ordered lists of numbers. A sequence is a list of numbers in a specific order, while a series is the sum of a sequence of numbers.

In this exercise, the problem involves identifying the next term in a sequence defined by a recurrence relation: \(P_k = \frac{5}{k(k+1)}\). Recurrence relations are equations that express each term of the sequence in relation to its predecessors.

When working with sequences and series:

• Identify any patterns or rules that define the sequence.
• Use these rules to find additional terms or to describe the general form of the sequence.
• For series, determine if the sum of terms leads to a closed form or if it converges to a particular value.

Understanding sequences and series is crucial for analyzing patterns and behavior in various mathematical contexts, making them a valuable tool in solving complex problems.