Question

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

Short answer

The supposed closed-form \( b_n = 2 \cdot 3^n \) does not match provided base cases \( p_0 = 1 \), \( p_1 = 2 \). Re-evaluate sequence properties for validity.

Step-by-step solution

Understand the Problem

We are given a recursive sequence defined by \( p_0 = 1 \), \( p_1 = 2 \), and \( p_n = 2p_{n-1} - p_{n-2} \) for \( n \geq 2 \). We need to prove that the closed form \( b_n = 2 \cdot 3^n \) fits this sequence.

Establish Base Cases

For the base case: Check \( n = 0 \) and \( n = 1 \).- \( b_0 = 2 \cdot 3^0 = 2 \cdot 1 = 2 \) not equal to \( p_0 = 1 \). It seems there is an error in the candidate formula because this must equal \( p_0 \).- \( b_1 = 2 \cdot 3^1 = 6 \) not equal to \( p_1 = 2 \). So we need a re-evaluation of the proposed closed form or initial conditions.

Analyze Inductive Hypothesis

Assuming \( b_k = 2 \cdot 3^k \) holds for some arbitrary \( k \geq 0 \), we look to prove it for \( k+1 \). However, since the base cases already demonstrate a potential issue, re-evaluate the sequence definition or proposed closed form.

Correct Error

Given the discrepancy already noticed between the supposed closed form and the initial conditions, it seems likely \( b_n = 2 \cdot 3^n \) is incorrectly framed. The correct closed form must satisfy \( b_0 = 1 \) and \( b_1 = 2 \).

Finding Correct Form

For correctly assessing \( b_n \) form consider:- Set \( b_0 = 1 \) and \( b_1 = 2 \) corresponding to initial given terms.- This suggests we redefine closed-form hypotheses or re-evaluate sequence defining details.

Key concepts

Closed Form Solution

A closed form solution is an expression that allows us to directly compute any term in a sequence without having to refer back to the previous terms. This is especially useful when dealing with recursive sequences, which often define terms based on prior terms. In our exercise, we're dealing with the sequence that is recursively defined as:

• \( p_0 = 1 \)
• \( p_1 = 2 \)
• \( p_n = 2p_{n-1} - p_{n-2} \) for \( n \geq 2 \)

The goal is to find a closed form for this sequence, initially assumed to be \( b_n = 2 \cdot 3^n \). However, upon checking against the base cases, there is a mismatch. This means a different closed form needs to be found that satisfies the conditions \( b_0 = 1 \) and \( b_1 = 2 \). Once a suitable form is found, the equation should enable calculation of any term directly, avoiding iterative calculations through previous sequence terms.

Inductive Step

The inductive step is crucial in proving that a closed form solution is valid for all terms in a sequence. It operates under the principle of mathematical induction, which has two key parts: the base case and the inductive step itself.
In our scenario, to prove that a form like \( b_n = 2 \cdot 3^n \) holds for all \( n \), assuming that it somehow satisfies its initial conditions (which it doesn’t, highlighting a need for a correct form), an inductive step is used.
Assuming the expression holds true for any arbitrary term \( k \), i.e., \( b_k = 2 \cdot 3^k \), we employ this to prove its validity for \( k+1 \):

• Show that if the given formula is true for \( k \), then it must also be true for \( k+1 \).
• Typically involves algebraically manipulating \( b_k \) to derive \( b_{k+1} \), thus confirming its consistency across the sequence.

However, due to the discrepancies with the base cases in our exercise, this step calls for re-assessment of the initially assumed closed form, as proving a correct form hinges on initial condition alignment.

Base Case

Establishing the base case is the first essential step in the process of mathematical induction. It serves as the foundational proof that the closed form solution -- if any -- applies to the simplest or smallest version of the problem.
In the case of the sequence's initially proposed closed form \( b_n = 2 \cdot 3^n \), we test against the given recursive sequence values:

• For \( n = 0 \), the closed form gives \( b_0 = 2 \cdot 3^0 = 2 \), whereas the recursive definition states \( p_0 = 1 \)
• For \( n = 1 \), the closed form gives \( b_1 = 2 \cdot 3^1 = 6 \), whereas the recursive definition states \( p_1 = 2 \)

The mismatch highlights that \( b_n = 2 \cdot 3^n \) does not satisfy the initial conditions of the sequence, proving the necessity to revise the closed form or clarify the conditions under which it holds true. Establishing a correct base case is crucial as it confirms the reliability of an inductive proof strategy moving forward in the sequence.