Problem 8 - Precalculus with Limits

Question

In Exercises 5 - 10, find \( P_{k + 1} \) for the given \( P_k \). \( P_k = \dfrac{k}{3} \left(2k + 1\right) \)

Step-by-Step Solution

Answer

The expression for \(P_{k+1}\) is \(P_{k + 1} = \dfrac{k + 1}{3} \left(2k + 3\right)\)

Step 1: Understand the given expression

The given expression is \(P_k = \dfrac{k}{3} \left(2k + 1\right)\). This is the value of \(P_k\).

Step 2: Replace \(k\) with \(k + 1\)

Substitute every occurrence of \(k\) in the given equation with \(k+1\). This will give us the expression for \(P_{k + 1}\). So, \(P_{k + 1} = \dfrac{k + 1}{3} \left(2 \cdot (k + 1) + 1\right)\).

Step 3: Simplify the expression

After substituting, simplify the obtained equation. So, \(P_{k + 1} = \dfrac{k + 1}{3} \left(2k + 2 + 1\right) = \dfrac{k + 1}{3} \left(2k + 3\right)\)

Key Concepts

Mathematical Induction

Mathematics often refers to mathematical induction as a method used to prove statements, particularly those about sequences or sums. The basic idea is simple but powerful: if you can prove a statement is true for a starting point and also show that if it's true for one stage it's true for the next, then it must be true for all stages.

• **Base Case**: You start by verifying the statement for an initial value, often called the base case. This is fundamental since it ensures a starting point for the truth of the statement.
• **Inductive Step**: Assume the statement is true for some arbitrary step, say for a general term, and then prove it for the following term. Essentially, you're establishing a domino effect where, if one domino falls (the base case), and every domino knocks over the next (the inductive step), all will eventually fall.

Mathematical induction is essentially a chain reaction, starting at a base case and continuing indefinitely. It’s particularly useful in showing properties about sequences or sums are true for all natural numbers.

Recursive Sequences

Recursive sequences formulate each term as a function of its preceding terms. It builds a bridge to the future using the past. Unlike explicit formulas, which give the nth term directly, recursive formulas need a few terms to generate the sequence.

Recursive relations are defined by providing:

• **Base Values**: These are the known values often present at the start of the sequence, which help compute future terms.
• **Recursion Rule**: A rule or formula that describes the relationship between successive terms in the sequence.

In our context of finding \( P_{k+1} \) from \( P_k \), we are using a recursion rule. Here we adjust \( P_k \) by incrementing \( k \) and applying the formula anew. Recursive sequences can appear complex at first, but when solved step-by-step, they develop into a flow that guides itself through its inherent rules.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions to make them more understandable or to solve equations. In this exercise, simplifying the expression after substituting k + 1 involved basic algebraic laws.
Some key techniques include:

• **Substitution**: Replacing variables with their expressions. For instance, replacing \( k \) with \( k+1 \).
• **Expansion**: Distributing values over expression terms, as seen when multiplying \( (k+1) \) with \( (2k + 3) \).
• **Grouping and Combining like terms**: This involves gathering similar terms to simplify the expression further.

In our example, after applying these techniques to \( P_{k+1} = \frac{(k + 1)}{3} (2k + 3) \), we moved from substitution to expansion, expressing the equation in a form fitting for further calculations or simplifications.