Problem 105 - Essential Calculus

Question

Let \(\left\\{x_{n}\right\\}, n \geq 0,\) be a sequence of nonzero real numbers such that \(x_{n}^{2}-x_{n-1} x_{n+1}=1\) for \(n=1,2,3, \ldots .\) Prove that there exists a real number \(a\) such that \(x_{n+1}=a x_{n}-x_{n-1},\) for all \(n \geq 1 .\)

Step-by-Step Solution

Answer

For the sequence of numbers \(x_{n}\) satisfying the condition \(x_{n}^{2}-x_{n-1} x_{n+1}=1\), there exists a real number \(a\) such that \(x_{n+1}=a x_{n}-x_{n-1}\), where \(a\) is obtained by \(a = \sqrt{a_{n}/a_{n-1}}\).

Step 1: Applying the given condition

The first stepping stone to our solution is by applying the conditional relation given in the problem \(x_{n}^{2}-x_{n-1} x_{n+1}=1\) for all \(n=1,2,3,\ldots .\) This equation itself gives us a sequence of the form \(x_{n}= a(x_{n-1}) -x_{n-2}\) where \(x_{0} = a(x_{-1}) - x_{-2}\) and \( x_{1} = a{x_{0}} - x_{-1}\). This can then help us establish our final required sequence. Thus we start with these two relations.

Step 2: Finding the value of a

For all \(n =1,2,3,\ldots,\) given \(x_{n}^{2}-x_{n-1} x_{n+1}=1\), we add \(x_{n-1}x_{n+1}\) to both sides of the equation to isolate \(x_{n}^{2}\) on LHS. Now we get \(x_{n}^{2} = 1 + x_{n-1} x_{n+1}\). We will represent the R.H.S of the equation as \(a_{n}\), so \(a_{n} = 1 + x_{n-1} x_{n+1}\). Which means, \(x_{n} = \sqrt{a_{n}}\). Similarly performing the same action for n-1, we get \(x_{n-1} = \sqrt{a_{n-1}}\). Dividing the two equations, we have \(x_{n}/x_{n-1} = \sqrt{a_{n}/a_{n-1}}\), solving which we get \(a = \sqrt{a_{n}/a_{n-1}}\)

Step 3: Forming the final sequence

Now, we introduce the final sequence for proof. We can write this original equation and the equation that we started with in Step 1, which is \(x_{n+1} = a_{n+1}x_{n} - x_{n-1}\). We then plug the value of a that we just obtained into this equation. This means, for \(n \geq 1 \), \(x_{n+1} = \sqrt{a_{n+1}/a_{n}} \cdot x_{n} - x_{n-1}\), which is the desired sequence equation, and it holds true for all real number sequences satisfying the original condition \(x_{n}^{2} - x_{n-1}x_{n+1} = 1\) for all \(n=1,2,3,\ldots .\)

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique that allows us to prove a statement or property for all natural numbers. It is akin to a domino effect; you prove the statement for the first natural number (usually starting at 1 or 0), and then show that, if the statement holds true for some arbitrary natural number, it must also hold true for the next natural number.

Applying mathematical induction to the sequence exercise, one would start by checking the base case—often the initial term of the sequence, labeled as x_0 or x_1, depending on the sequence's definition. After verifying that the base case satisfies the given property, we then assume that the property holds for some arbitrary term x_n, and using this assumption, we prove that the property must hold for x_{n+1}. This step is the inductive step.

If both the base case and the inductive step are proven, we've shown via mathematical induction that the statement or property holds across all terms of the sequence.

Recurrence Relationships

Recurrence relationships, also known as recurrence relations, are equations that define sequences of values based on previous terms in the sequence. They're fundamental in disentangling complex systems by expressing subsequent values as functions of their predecessors.

In the context of the provided exercise, the recurrence relationship reveals how every term of the sequence (x_n) is related to one another. The condition x_{n}^{2} - x_{n-1} x_{n+1} = 1 implicitly describes a recurrence because it connects a term with its surrounding neighbors.

Understanding recurrence relationships leads to an iterative method for constructing the entire sequence, provided the first few terms. They also pave the way for detection of patterns within sequences and finding explicit formulas or at least simplifying the terms of the sequence for easier calculation and analysis.

Real Number Properties

The properties of real numbers are the bedrock on which many mathematical concepts are built. Primarily, there are four key properties: Commutative, Associative, Distributive, and the existence of Identity elements.

The Commutative property states that a + b = b + a and ab = ba for all a, b in real numbers, meaning addition and multiplication can be performed in any order. The Associative property allows us to group numbers without changing the result, that is (a+b)+c = a+(b+c) and (ab)c = a(bc).

The Distributive property bridges addition and multiplication, stating a(b + c) = ab + ac. Lastly, the Identity properties refer to the existence of an additive identity, 0, and a multiplicative identity, 1, for which a + 0 = a and a * 1 = a for any real number a.

In the exercise given, the real number properties ensure that sequences can be manipulated algebraically to reach the desired form x_{n+1} = a x_{n} - x_{n-1}, leveraging the real number properties to simplify and solve the sequence relationship.