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Magazine Chapter 9: Problem 106
Assume that each sequence converges and find its limit. $$a_{1}=3, \quad a_{n+1}=12-\sqrt{a_{n}}$$
Short Answer
Expert verified
The limit of the sequence is 9.
Step by step solution
01
Understand the Concept of Convergence
When a sequence converges, it means it approaches a certain value, the limit, as the index goes to infinity. We will denote this limit by \( L \).
02
Set the Limit Equation
Assume that the sequence \( \{a_n\} \) converges to \( L \). Then the limit of the sequence \( a_{n+1} \) should also be \( L \). Therefore, set up the equation: \( L = 12 - \sqrt{L} \).
03
Solve the Limit Equation
Rearrange the equation \( L = 12 - \sqrt{L} \) to isolate \( \sqrt{L} \). This gives: \( \sqrt{L} = 12 - L \). Squaring both sides results in the equation \( L = (12 - L)^2 \).
04
Expand and Rearrange the Equation
Expand the right-hand side: \( L = 144 - 24L + L^2 \). Bring all terms to one side to form a quadratic equation: \( L^2 - 25L + 144 = 0 \).
05
Solve the Quadratic Equation
Use the quadratic formula \( L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -25 \), and \( c = 144 \). Calculate the discriminant \( b^2 - 4ac = 625 - 576 = 49 \).
06
Find the Roots
Substitute into the quadratic formula: \( L = \frac{25 \pm \sqrt{49}}{2} \). The roots are \( L = \frac{25 + 7}{2} = 16 \) and \( L = \frac{25 - 7}{2} = 9 \).
07
Determine the Valid Limit
Check the behavior of the sequence and initial conditions. With \( a_1 = 3 \), both 9 and 16 are potential limits. Check \( a_2 = 12 - \sqrt{3} \approx 10.268 \), which suggests the sequence is increasing and heads towards 9.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limit of a Sequence
In calculus and mathematical analysis, understanding the limit of a sequence is crucial. A sequence is simply an ordered list of numbers, and when it converges, it approaches a certain value as you look further down the list. This value that the sequence approaches is called the limit.
For example, let's say we have a sequence \( \{a_n\} \) where \( a_1 = 3 \) and \( a_{n+1}=12-\sqrt{a_n} \). We assume this sequence converges, meaning it gets closer and closer to a specific number, denoted as \( L \), as \( n \) goes to infinity.
For example, let's say we have a sequence \( \{a_n\} \) where \( a_1 = 3 \) and \( a_{n+1}=12-\sqrt{a_n} \). We assume this sequence converges, meaning it gets closer and closer to a specific number, denoted as \( L \), as \( n \) goes to infinity.
- The concept of a limit is foundational to calculus as it helps define continuity, derivatives, and integrals.
- The sequence must behave nicely, meaning it should neither oscillate wildly nor grow without bound, for a limit to exist.
- Finding the limit involves setting up an equation based on the sequence鈥檚 behavior and solving it, as we saw with \( L = 12 - \sqrt{L} \).
Quadratic Equation
In the study of sequences, quadratic equations often pop up, especially when finding limits. A quadratic equation is any equation that can be represented in the form \( ax^2 + bx + c = 0 \). The solution to this can be found using the quadratic formula:
\[ L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our sequence example, after setting up and rearranging the limit equation, we find ourselves with a quadratic equation: \( L^2 - 25L + 144 = 0 \).
Understanding the use of the quadratic formula allows us to solve polynomial equations that arise when studying sequences.
\[ L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our sequence example, after setting up and rearranging the limit equation, we find ourselves with a quadratic equation: \( L^2 - 25L + 144 = 0 \).
- To solve this, we identify \( a = 1 \), \( b = -25 \), and \( c = 144 \).
- The discriminant \( b^2 - 4ac \) helps determine the nature of the solutions; a positive discriminant indicates two real and distinct solutions.
- Using the quadratic formula, we find the roots \( L = 16 \) and \( L = 9 \).
Understanding the use of the quadratic formula allows us to solve polynomial equations that arise when studying sequences.
Convergence in Calculus
Convergence is a key concept in calculus, essential for understanding how functions and sequences behave over time. When a sequence converges, its terms get closer and closer to a specific value, known as its limit. This behavior is important as it helps mathematicians analyze complex systems.
For our sequence \( \{a_n\} \) with \( a_1 = 3 \) and \( a_{n+1}=12-\sqrt{a_n} \), we ultimately found the sequence converges to a limit of 9, though initially we determined two possible limits, 9 and 16. The reason it converges specifically to 9 can be identified by analyzing the behavior of the sequence:
For our sequence \( \{a_n\} \) with \( a_1 = 3 \) and \( a_{n+1}=12-\sqrt{a_n} \), we ultimately found the sequence converges to a limit of 9, though initially we determined two possible limits, 9 and 16. The reason it converges specifically to 9 can be identified by analyzing the behavior of the sequence:
- The sequence's starting value and recursive formula determine the path and progression of numbers.
- In practice, testing initial numbers can help verify which of the possible limits is viable.
- Understanding convergence provides insights into how mathematical systems stabilize and is crucial for further studies of continuity and differentiability.
