Question

The terms of a sequence are given recursively as \(p_{0}=3, p_{1}=7,\) and \(p_{n}=3 p_{n-1}-\) \(2 p_{n-2}\) for \(n \geq 2\). Write out the information that the inductive step assumes and what the step must prove in proving \(b_{n}=2^{n+2}-1\) is a closed form for the sequence. Suppose \(n_{0}=1\) and the base cases are 0 and \(1 .\)

Short answer

Verify base cases and assume the hypothesis for inductive step.

Step-by-step solution

Base Case Verification

First, we need to check the base cases for \( n = 0 \) and \( n = 1 \). We will verify whether \( b_n = 2^{n+2} - 1 \) holds true for these initial values. For \( n = 0 \): \[ b_0 = 2^{0+2} - 1 = 2^2 - 1 = 4 - 1 = 3 \]This matches \( p_0 = 3 \).For \( n = 1 \):\[ b_1 = 2^{1+2} - 1 = 2^3 - 1 = 8 - 1 = 7 \]This matches \( p_1 = 7 \). Since both base cases hold true, we can proceed to the inductive step.

Inductive Hypothesis

Assume that the closed form \( b_n = 2^{n+2} - 1 \) holds true for some arbitrary integer \( n = k \) and \( n = k - 1 \). That is:1. \( p_k = 2^{k+2} - 1 \)2. \( p_{k-1} = 2^{k+1} - 1 \) These assumptions are the basis from which we'll prove that \( b_n \) holds for \( n = k+1 \).

Key concepts

Recursive Sequences

Recursive sequences are a fascinating aspect of mathematics that involves defining each term of a sequence based on previous terms. In our original exercise, the sequence is defined with an explicit recursion: \( p_{n} = 3p_{n-1} - 2p_{n-2} \) along with base cases \( p_0 = 3 \) and \( p_1 = 7 \). This recursive formula allows each new term \( p_n \) to depend on its two preceding terms \( p_{n-1} \) and \( p_{n-2} \).
Recursive sequences are used to break complex problems into simpler forms, using known initial conditions to build the sequence step by step. To work with these sequences, you calculate the first few terms using the base cases, and then use the recursive relationship to find additional terms. In this way, recursive sequences offer a dynamic way of defining terms when a direct formula is not available. Understanding the way each term connects helps in leveraging these relationships to find patterns or potential closed forms as shown in the exercise.

Closed Form Expression

A closed form expression is an equation that allows you to calculate any term in a sequence directly, without needing to compute all the previous terms. In the original problem, the formula \( b_n = 2^{n+2} - 1 \) is presented as a closed form for the recursively defined sequence. With closed forms, you replace recursion with a clear mathematical equation that is often simpler to work with.
Finding a closed form can be quite beneficial because it simplifies computations and enables quick predictions of term values. Even though recursive definitions are concise, they may become cumbersome for large \( n \). In contrast, a closed form like \( 2^{n+2} - 1 \) can be calculated easily for any value of \( n \) without iterative calculations. This means you could, for instance, compute the 1000th term with as much ease as the 5th.

Base Case Verification

Base case verification is an essential step when dealing with recursive sequences or proving closed forms. It's akin to establishing a strong foundation for a building. In our solution, we start by confirming that the given closed form \( b_n = 2^{n+2} - 1 \) holds true for the initial terms \( n = 0 \) and \( n = 1 \).
- For \( n = 0 \), substituting into the closed form, we find \( b_0 = 3 \), which matches our base case \( p_0 = 3 \).- For \( n = 1 \), similarly, substituting gives \( b_1 = 7 \), which matches \( p_1 = 7 \).
Verification of base cases is crucial because it ensures that your expression is initially aligned with the known values of the sequence. It gives confidence that the formula is set correctly and can be trusted as a basis for further inductive reasoning.

Mathematical Induction

Mathematical induction is a powerful technique used to prove the truth of an infinite number of cases, often used with sequences. After verifying the base cases, we move into the induction step. We assume that the statement \( b_n = 2^{n+2} - 1 \) holds for an arbitrary value \( n = k \), often called the "inductive hypothesis".
- Assume it's true for \( n = k \) and \( n = k-1 \): - \( p_k = 2^{k+2} - 1 \) - \( p_{k-1} = 2^{k+1} - 1 \)The next step is critical: use these assumptions to prove that the formula holds for \( n = k+1 \). If you can successfully show this, by reasoning through the recursive relationship and your hypothesis, you conclude that the formula works for all cases. This method, akin to falling dominoes, ensures that if it's true for one, and the transition from one step to the next is valid, it will be true for all. Inductive reasoning in mathematics walks you from certainty in known cases to validity in all future cases.