Problem 9 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The given recursive sequence \( P_{k + 1} \) can be represented as \( \dfrac{3}{(k + 3)(k + 4)} \)

Step 1: Understand the problem

It is important to understand what the problem is asking. We are given a sequence \( P_k \) that is defined recursively, meaning the value of \( P_k \) depends on the index \( k \). We are asked to find an expression for \( P_{k + 1} \) in terms of \( k \).

Step 2: Shift the indices in the formula

Since we are looking for \( P_{k + 1} \), we need to replace every \( k \) in the expression for \( P_k \) with \( k + 1 \) in order to shift the indices. So, \( P_{k + 1} = \dfrac{3}{\left((k + 1) + 2\right)\left((k + 1) + 3\right)} = \dfrac{3}{(k + 3)(k + 4)} \)

Step 3: Simplifying the expression

There is no real need to simplify the expression but it's always a good step to check. So, after simplification, our new sequence \( P_{k + 1} \) is \( \dfrac{3}{(k + 3)(k + 4)} \)

Key Concepts

Index Shifting

When dealing with recursive sequences, one frequent task is index shifting. This means adjusting the indices in a sequence formula to find a subsequent term in the series. In this particular exercise, given the sequence formula for \( P_k \), we are tasked to find \( P_{k + 1} \) by shifting the index from \( k \) to \( k+1 \).

To perform index shifting, we replace each occurrence of \( k \) in the original formula with \( k+1 \). For example, if the original term is \( P_k = \frac{3}{(k+2)(k+3)} \), then \( P_{k+1} \) becomes \( \frac{3}{((k+1)+2)((k+1)+3)} \).

This method ensures that the sequence retains its rules as it progresses, allowing us to consistently calculate subsequent terms by following a systematic approach.

Sequence Formula

The sequence formula provides a mathematical representation of elements within a sequence. It helps define how terms are related to their predecessors. In recursive sequences, each term is specifically defined based on the previous ones, characterized by \( P_k \) here.

In this exercise, our sequence is defined through \( P_k = \frac{3}{(k+2)(k+3)} \). This formula shows the relationship of \( P_k \) through denominators \( (k+2) \) and \( (k+3) \), showcasing how each term alters as \( k \) changes. Understanding the sequence formula also aids in recognizing patterns or rules within the series.

For recursive sequences like this, finding the next term involves adhering to the given formula while shifting indices appropriately, leading us efficiently to \( P_{k+1} = \frac{3}{(k+3)(k+4)} \).

Simplifying Expressions

Simplification in math refers to reducing expressions to their simplest form. While the expression for the new sequence term \( P_{k+1} = \frac{3}{(k+3)(k+4)} \) doesn't require much further simplification, it's always important to ensure that the result is concise and no minor simplifications are overlooked.

Simplifying might involve cancelling common factors, reducing complex fractions, or occasionally factoring polynomials. While not always necessary, especially in basic expressions, checking your work for any potential simplifications ensures accuracy and clarity in communication of mathematical ideas.

By consistently applying simplifying techniques, students can develop keen analytical skills that streamline the problem-solving process and lead to more elegant solutions in other math problems.