Problem 10 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

\( P_{k + 1} = \frac{k^2 + 2k + 1}{2(k^2 + 4k + 4)} \)

Step 1: Substitute k with (k+1)

The first step is to substitute \( k \) with \( k + 1 \) in the formula for \( P_k \). So \( P_{k + 1} \) is \( \frac{(k + 1)^2}{2 ((k + 1) + 1)^2} \).

Step 2: Simplify the formula

Next, we simplify the formula. In the numerator, \( (k + 1)^2 \) becomes \( k^2 + 2k + 1 \). In the denominator, \( 2((k + 1) + 1)^2 \) becomes \( 2(k + 2)^2 \) which then simplifies to \( 2(k^2 + 4k + 4) \). Thus, \( P_{k + 1} = \frac{k^2 + 2k + 1}{2(k^2 + 4k + 4)} \).

Key Concepts

Mathematical Induction

Understanding mathematical induction is vital for diving into more complex mathematical concepts. It is a method of proof used to establish that a given statement is true for all positive integers. The process begins with checking the statement for the initial case, often when the value is 1. Then, the inductive step involves assuming the statement is true for some integer k (the inductive hypothesis) and proving it is also true for the next integer, k+1.

The exercise at hand is an example where mathematical induction could be used to prove a property of the sequence defined recursively by a formula for any positive integer k. However, the exercise focuses only on finding the next term in the sequence, which is a common step within a proof by induction. When applying induction, students should remember to clearly outline each step: the base case, the inductive hypothesis, and the inductive step.

Sequences and Series

Sequences and series are fundamental concepts in mathematics, particularly in algebra and calculus. A sequence is a list of numbers in a particular order, and each number in the sequence is called a term. For instance, the sequence provided in the exercise, defined by the recursive formula \( P_k = \frac{k^2}{2(k + 1)^2} \) given for a positive integer k, produces a series of terms when k is varied.

To deepen understanding, exploring the relationship between consecutive terms is key. By finding \( P_{k + 1} \) as in the step-by-step solution, students practice working with sequences. They also learn how the terms of a sequence are generated and how patterns in the sequence evolve. Recognizing and generalizing these patterns can be critical for solving more complex problems involving sequences and series.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. A robust understanding of how to manipulate these expressions is essential for solving a wide array of problems in algebra. The solution of the exercise exemplifies simplifying a complex algebraic expression.

During the simplification process, expanding \( (k + 1)^2 \) to \( k^2 + 2k + 1 \) and \( (k + 2)^2 \) to \( k^2 + 4k + 4 \) demonstrates how to handle binomial expansion. Moreover, recognizing common terms and reducing fractions are essential skills for working efficiently with algebraic expressions. By practicing problems like this, students become adept at manipulating expressions and can apply these skills across various areas of mathematics.