Questions: (b) Let (an=fracfn+1fn). Which of the following shows that (an-1=1+frac1an-2) ? (an=fracfn+1fn Rightarrow an-1=fracfnfn-1=fracfn-1+fn-2fn-1=1+fracfn-2fn-1=1+frac1fn-1 / fn-2=1+frac1an-2) (an=fracfn+1fn Rightarrow an-1=fracfnfn-1=fracfn+fn-1fn-1=1+fracfnfn-1=1+frac1fn-1 / fn=1+frac1an-2) (an=fracfn+1fn Rightarrow an-1=fracfnfn-2=fracfn-1+fn-2fn-2=1+fracfn-1fn-2=1+frac1fn-2 / fn-1=1+frac1an-2) (an=fracfn+1fn Rightarrow an-1=fracfn-1fn=fracfn-1fn-1+fn-2=1+fracfn-1fn-2=1+frac1fn-2 / fn-1=1+frac1an-2) (an=fracfn+1fn Rightarrow an-1=fracfn+2fn+1=fracfn+1+fnfn+1=1+fracfnfn+1=1+frac1fn / fn+1=1+frac1an-2) Assuming that (leftanright) is convergent, find its limit. (If an answer does not exist, enter DNE.)

(b) Let (an=fracfn+1fn). Which of the following shows that (an-1=1+frac1an-2) ? (an=fracfn+1fn Rightarrow an-1=fracfnfn-1=fracfn-1+fn-2fn-1=1+fracfn-2fn-1=1+frac1fn-1 / fn-2=1+frac1an-2) (an=fracfn+1fn Rightarrow an-1=fracfnfn-1=fracfn+fn-1fn-1=1+fracfnfn-1=1+frac1fn-1 / fn=1+frac1an-2) (an=fracfn+1fn Rightarrow an-1=fracfnfn-2=fracfn-1+fn-2fn-2=1+fracfn-1fn-2=1+frac1fn-2 / fn-1=1+frac1an-2) (an=fracfn+1fn Rightarrow an-1=fracfn-1fn=fracfn-1fn-1+fn-2=1+fracfn-1fn-2=1+frac1fn-2 / fn-1=1+frac1an-2) (an=fracfn+1fn Rightarrow an-1=fracfn+2fn+1=fracfn+1+fnfn+1=1+fracfnfn+1=1+frac1fn / fn+1=1+frac1an-2) Assuming that (leftanright) is convergent, find its limit. (If an answer does not exist, enter DNE.)
Transcript text: (b) Let $a_{n}=\frac{f_{n+1}}{f_{n}}$. Which of the following shows that $a_{n-1}=1+\frac{1}{a_{n-2}}$ ? $a_{n}=\frac{f_{n+1}}{f_{n}} \Rightarrow a_{n-1}=\frac{f_{n}}{f_{n-1}}=\frac{f_{n-1}+f_{n-2}}{f_{n-1}}=1+\frac{f_{n-2}}{f_{n-1}}=1+\frac{1}{f_{n-1} / f_{n-2}}=1+$ $\frac{1}{a_{n-2}}$ $a_{n}=\frac{f_{n+1}}{f_{n}} \Rightarrow a_{n-1}=\frac{f_{n}}{f_{n-1}}=\frac{f_{n}+f_{n-1}}{f_{n-1}}=1+\frac{f_{n}}{f_{n-1}}=1+\frac{1}{f_{n-1} / f_{n}}=1+\frac{1}{a_{n-2}}$ $a_{n}=\frac{f_{n+1}}{f_{n}} \Rightarrow a_{n-1}=\frac{f_{n}}{f_{n-2}}=\frac{f_{n-1}+f_{n-2}}{f_{n-2}}=1+\frac{f_{n-1}}{f_{n-2}}=1+\frac{1}{f_{n-2} / f_{n-1}}=1+$ $\frac{1}{a_{n-2}}$ $a_{n}=\frac{f_{n+1}}{f_{n}} \Rightarrow a_{n-1}=\frac{f_{n-1}}{f_{n}}=\frac{f_{n-1}}{f_{n-1}+f_{n-2}}=1+\frac{f_{n-1}}{f_{n-2}}=1+\frac{1}{f_{n-2} / f_{n-1}}=1+$ $\frac{1}{a_{n}-2}$ $a_{n}=\frac{f_{n+1}}{f_{n}} \Rightarrow a_{n-1}=\frac{f_{n+2}}{f_{n+1}}=\frac{f_{n+1}+f_{n}}{f_{n+1}}=1+\frac{f_{n}}{f_{n+1}}=1+\frac{1}{f_{n} / f_{n+1}}=1+\frac{1}{a_{n-2}}$ Assuming that $\left\{a_{n}\right\}$ is convergent, find its limit. (If an answer does not exist, enter DNE.) $\square$ Need Help? Read It
failed

Solution

failed
failed

Solution Steps

To find the limit of the sequence \( \{a_n\} \) given that it is convergent, we start by noting the relationship \( a_{n-1} = 1 + \frac{1}{a_{n-2}} \). If the sequence converges to a limit \( L \), then as \( n \) approaches infinity, \( a_n \), \( a_{n-1} \), and \( a_{n-2} \) all approach \( L \). Therefore, we can set up the equation \( L = 1 + \frac{1}{L} \) and solve for \( L \).

Step 1: Set Up the Equation

Given the relationship \( a_{n-1} = 1 + \frac{1}{a_{n-2}} \), we assume the sequence \( \{a_n\} \) converges to a limit \( L \). Therefore, we set up the equation: \[ L = 1 + \frac{1}{L} \]

Step 2: Solve the Equation

To find the limit \( L \), we solve the equation: \[ L = 1 + \frac{1}{L} \] Multiplying both sides by \( L \) to clear the fraction, we get: \[ L^2 = L + 1 \] Rearranging the equation, we obtain a quadratic equation: \[ L^2 - L - 1 = 0 \]

Step 3: Find the Roots of the Quadratic Equation

Solving the quadratic equation \( L^2 - L - 1 = 0 \) using the quadratic formula \( L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -1 \), and \( c = -1 \), we get: \[ L = \frac{1 \pm \sqrt{5}}{2} \]

Step 4: Determine the Convergent Limit

The roots of the quadratic equation are: \[ L = \frac{1 + \sqrt{5}}{2} \quad \text{and} \quad L = \frac{1 - \sqrt{5}}{2} \] Since \( \frac{1 - \sqrt{5}}{2} \) is negative and the sequence \( \{a_n\} \) consists of positive terms, the limit must be: \[ L = \frac{1 + \sqrt{5}}{2} \]

Final Answer

The limit of the sequence \( \{a_n\} \) is: \[ \boxed{L = \frac{1 + \sqrt{5}}{2}} \]

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Suppose a special 35-sided die is rolled once where each face is marked with a number 1 through 35. What is the probability that the die lands on a 4 or a 13?
failedAnswered
A new stadium will have 7,062 seats. The stadium designer wants to divide the seats into sections. Each section must have the same number of seats. How many sections could the stadium designer use? Select all the possible numbers. 10 9 6 5
failedAnswered
In the best of all worlds scientists would use to evaluate hypotheses. the comparative method nonparametric statistics experiments parametric statistics
failedAnswered
In a study of helicopter usage and patient survival, among the 47,773 patients transported by helicopter, 260 of them left the treatment center against medical advice, and the other 47,513 did not leave against medical advice. If 40 of the subjects transported by helicopter are randomly selected without replacement, what is the probability that none of them left the treatment center against medical advice?
failedAnswered
Practice Exercise The KP for the reaction N2(g) + 3 H2(g) ⇌ 2 NH3(g) is 4.3 × 10^-4 at 375°C. In one experiment, the initial pressures are: PH2 = 0.40 atm, PN1 = 0.86 atm, and PNH3 = 4.4 × 10^-3 atm. Calculate ΔG for the reaction and predict the direction of the net reaction.