Problem 6 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

\( P_{k+1} = \dfrac{1}{2(k+3)} \)

Step 1: Substitute \( k+1 \) into the formula

Substitute \( k+1 \) into the k in \( P_k = \dfrac{1}{2(k + 2)} \). This substitution gives \( P_{k+1} = \dfrac{1}{2((k+1) + 2)} \).

Step 2: Simplify the formula

Simplify the expression inside the parentheses and find the new value of \( P_{k+1} \). We obtain \( P_{k+1} = \dfrac{1}{2(k+3)} \).

Key Concepts

Sequence Formula

A sequence formula is a mathematical expression that defines a sequence of numbers. In the context of recursive sequences, each term in the sequence depends on the previous term or terms. Recursive sequences are defined using a rule or formula, making it easier to predict or calculate the next terms.

In our exercise, the sequence formula is given as:

• \( P_k = \dfrac{1}{2(k + 2)} \)

This formula tells us how each term \( P_k \) in the sequence can be computed using the index \( k \). As you can see, the formula uses the variable \( k \) to determine the value of the sequence at each position.

Understanding and defining the sequence formula correctly is essential, as it acts as the base for any further calculations or transformations such as substitution and simplification.

Substitution Method

The substitution method is a technique used to find specific terms in a sequence by replacing variables in the sequence formula with new values. This method is particularly useful when defining recursive sequences because it helps us move from one term to the next.

To find a new term in the sequence, say \( P_{k+1} \), you substitute \( k+1 \) into the sequence formula in place of \( k \). For this exercise:

• Original: \( P_k = \dfrac{1}{2(k + 2)} \)
• Substitute: \( k+1 \) in place of \( k \)
• Result: \( P_{k+1} = \dfrac{1}{2((k+1) + 2)} \)

The substitution method simplifies the process of extending sequences by converting known terms into the required next term's format. It highlights the dynamic aspect of recursive sequences where each term depends on a consistent rule.

Sequence Simplification

Sequence simplification is the process of making a sequence's terms easier to work with by reducing the complexity of its formula. This makes it simpler to identify patterns or calculate further terms.

In the given exercise, after substituting \( k+1 \) into the sequence formula, we arrived at:

• \( P_{k+1} = \dfrac{1}{2((k+1) + 2)} \)

To simplify:

• Calculate the expression in the parentheses: \((k + 1) + 2 = k + 3\)
• Update the formula to: \( P_{k+1} = \dfrac{1}{2(k+3)} \)

Simplification allows for more efficient calculations and a better understanding of the sequence's behavior. It helps to highlight any key features or patterns, making further exploration or computation smoother and more straightforward.