5. Using mathematical induction please. of them.Prove that for any ne N,Prove that for any nЄ N,3. Prove that for any nЄ N,12+32 +52 + ... + (2n − 1)² = n(2n − 1)(2n+1)1³ + 2³ + ... + n³ :~-I(+)1'34. Prove that for any nЄ N,2n1++1-(n + 1)!(n+1)!"2232n+≤2-115. The Fibonacci sequence is defined as follows: F₁ = F2 = 1, and for each n ≥ 3, F = F-1 + Fn-2. Thus, thefirst few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, .Prove that for any nЄ N, Fan-1 and F3n-2 are odd, while F3 is even.6. Consider a function f that is defined for all natural numbers as follows: f(1) = 1 and for each n ≥ 2,f(n) 3f(n-1)+ 10. Prove that for each natural number n, f(n) =2.3"-5.Y UBmsiF10F1 F12PG UPDELENDPG DN7HOMETNEW

5. To prove that F3n−1 and F3n−2 are odd while F3n is even for any n∈N, it can use mathematical…

Answered: 5. The Fibonacci sequence is defined as…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Recruiting In September 2002, CNN reported on amethod of grad student recruiting by the Haas School ofBusiness at U.C.-Berkeley. The school notifies applicantsby formal letter that they have been admitted, and alsoe-mails the accepted students a link to a Web site thatgreets them with personalized balloons, cheering, andapplause. The director of admissions says this extra effortat recruiting has really worked well. The school accepts500 applicants each year, and the percentage that actuallychoose to enroll at Berkeley increased from 52% the yearbefore the Web greeting to 54% this year.a) Create a 95% confidence interval for the change inenrollment rates.b) Based on your confidence interval, are you convincedthat this new form of recruiting has been effective?Explain.
The integers from 1 to 7 are written in order clockwise around a circle. I start with a counter on number 1 and repeatedly roll a standard six-sided fair die. If the die shows 1 then I move the counter to the next number clockwise around the circle; if the die shows 2 or 3 then I move the counter to the next number anticlockwise around the circle; if the die shows 4, 5 or 6 then I do not move the counter. (i) Describe how to model this process as a discrete-time Markov chain. Write the transition matrix (ii) Explain why the Markov property is satisfied and give an example of how you could make a small modification after which the process would be a discrete-time Markov chain which is not homogeneous.
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sed on the slot machine shown to the right, in how many ways can we obtain bars on all three whe nere are ways to obtain bars on all three wheels.
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A die is rolled two times Provide a list of all the possible outcomes of the two rolls in this form: the result from the first roll and results from the second roll.The answer needs to be done using R which I am confused on how to do
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5. The Fibonacci sequence is defined as follows: F₁ = F2 = 1, and for each n ≥ 3, F = F-1 + F-2. Thus, the first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34,. Prove that for any nЄ N, Fan-1 and Fan-2 are odd, while Fan is even.