Problem 91 - Thomas Calculus

Question

In Exercises \(91-98\) , assume that each sequence converges and find its limit. $$ a_{1}=2, \quad a_{n+1}=\frac{72}{1+a_{n}} $$

Step-by-Step Solution

Answer

The limit of the sequence is 8.

Step 1: Identify the Recursive Sequence

The sequence given is recursive, where each term is defined using the previous term. We are given the first term, \( a_1 = 2 \), and the relationship for subsequent terms is \( a_{n+1} = \frac{72}{1 + a_n} \).

Step 2: Assume a Limit Exists

Since the sequence is assumed to converge, let \( L \) be the limit of the sequence. Therefore, as \( n \rightarrow \infty \), both \( a_n \) and \( a_{n+1} \) approach \( L \).

Step 3: Substitute the Limit Into the Recursive Formula

If \( a_n \rightarrow L \), then the limit of the recursive formula becomes \( L = \frac{72}{1+L} \).

Step 4: Solve the Limit Equation

Rearrange the limit equation to find \( L \). Multiply both sides by \( 1+L \), resulting in:\[ L(1 + L) = 72 \]Expand and rearrange:\[ L^2 + L = 72 \]\[ L^2 + L - 72 = 0 \]Solve this quadratic equation using the quadratic formula \( L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 1 \), and \( c = -72 \).

Step 5: Calculate Using the Quadratic Formula

Substitute \( a = 1 \), \( b = 1 \), \( c = -72 \) into the quadratic formula:\[ L = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-72)}}{2 \cdot 1} \]\[ L = \frac{-1 \pm \sqrt{1 + 288}}{2} \]\[ L = \frac{-1 \pm \sqrt{289}}{2} \]\[ L = \frac{-1 \pm 17}{2} \]

Step 6: Determine the Real Solution for L

The possible solutions are:\[ L = \frac{16}{2} = 8 \]\[ L = \frac{-18}{2} = -9 \]Since the sequence starts with a positive number and the terms are positive, the converging limit must be positive. Therefore, \( L = 8 \).

Key Concepts

Recursive Sequence

A recursive sequence is a sequence of numbers where each term is derived from prior terms. In this context, we're given an initial value, \( a_1 = 2 \), and a formula that calculates the next term based on the previous term, \( a_{n+1} = \frac{72}{1 + a_n} \). This means that to find any term in the sequence, you perform a calculation that involves the previous term in the sequence.

Recursive sequences are fascinating because they establish a direct relationship between successive elements, creating a chain-like dependency.
Imagine a line of dominoes where each domino's fall depends on the one before it. Similarly, each term in a recursive sequence "falls" into place based on the value of the earlier term.

• They often require initial conditions, like our \( a_1 = 2 \), to begin the calculations.
• While they can seem complex, establishing a pattern often simplifies their analysis.

Limit of Sequence

The limit of a sequence is a concept that helps us understand the long-term behavior of sequences. When a sequence converges, its terms tend to approach a specific number, known as the limit. For example, in our exercise, as \( n \) becomes very large, the terms \( a_n \) approach the limit \( L \).

It's crucial to assume that a limit exists when dealing with convergence tasks. You hypothesize this because only converging sequences reach a fixed point over time. In the exercise, it’s acknowledged that \( a_n \rightarrow L \) as \( n \to \infty \).
Understanding limits doesn’t just give us an ending point for sequences; it helps predict the sequence's behavior without calculating many individual terms.

• To find the limit, substitute the limit into the recursive formula.
• This substitution allows you to solve for \( L \) using algebraic manipulation.

Hence, identifying the limit makes sequences much easier to manage and analyze.

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, generally in the form \( ax^2 + bx + c = 0 \). They often arise from problems involving limits and recursive sequences, as seen in the given exercise.

To find the limit \( L \), we ended up with a quadratic equation \( L^2 + L - 72 = 0 \). Solving this involved using the quadratic formula \( L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). The quadratic formula is a standard tool for finding the roots of any quadratic equation.

When solving quadratic equations:

• First, ensure the equation is in the form \( ax^2 + bx + c = 0 \).
• Identify the coefficients \( a \), \( b \), and \( c \).
• Substitute into the quadratic formula and compute \( L \).

In the solution, roots were calculated, and by analyzing the context, we selected the viable number \( L = 8 \) as it was positive. The quadratic equation here acts as a bridge to determine the limit of the sequence.