Can you fill in the blanks for me? Fill in the blanks in the following proof, which shows that the sequence defined by the recurrence relation+ 2* for each integer k 2 2= 1satisfies the following formula.f. = 2" +1- 3 for every integer n21Proof (by mathematical induction):Suppose f,, f,, f... is a sequence that satisfies the recurrence relationf, = f, + 2* for each integer k 2 2, with initial condition f, = 1.We need to show that when the sequence f,, f,, f,... is defined in this recursive way, all the terms in the sequence also satisfy the explicit formula shown above.So let the property P(n) be the equation f, = 2" +1- 3. We will show that P(n) is true for every integer nz 1.Show that P(1) is true:The left-hand side of P(1) is 22 – 3which equals 1The right-hand side of P(1) is 1v . Since the left-hand and right-hand sides equal each other, P(1) is true.Show that for each integer k21, if P(k) is true, then P(k + 1) is true:Let k be any integer with k z 1, and suppose that P(k) is true. In other words, suppose that f, = 2*+1- 3where f,, f2, f3,... is a sequence defined by the recurrence relationf = f -1+ 2* for each integer k z 2, with initial condition f, = 1.[This is the inductive hypothesis.]We must show that P(k + 1) is true. In other words, we must show that f1gt+2 - 3. Now the left-hand side of P(k + 1) isfe+1 = f, +2* + 1by definition of f, f, f+ 2k + 1by inductive hypothesisby the laws of algebraand this is the right-hand side of P(k + 1). Hence the inductive step is complete.[Thus, both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.] If a bank pays interest at a rate of i compounded m times a year, then the amount of money P, at the end of k time periods (where one time period = 1/mth of a year)satisfies the recurrence relation P, =k1 +Pk - 1with initial condition P.= the initial amount deposited. Find an explicit formula for P,n'mThe given recurrence relation defines ---Select---v with constant ---Select---which is. Therefore, Pnfor every%3Dinteger n 2 0.

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Fill in the blanks in the following proof, which shows that the sequence defined by the recurrence relation + 2* for each integer k z 2 - 1 atisfies the following formula. f- 2" +1- 3 for every integer n21 Proof (by mathematical induction): Suppose f,. f. fg. is a sequence that satisfies the recurrence relation f-f + 2" for each integer k z 2, with initial condition f,- 1. Ne need to show that when the sequence f,, f, fy... is defined in this recursive way, all the terms in the sequence also satisfy the exxplicit formula shown above. So let the property P(n) be the equation f- 2" +1 - 3. We will show that P(n) is true for every integer n2 1. Show that P(1) is true: The left-hand side of P(1) is 22 – 3 , which equals 1 . The right-hand side of P(1) is 1 . Since the left-hand and right-hand sides equal each other, P(1) is true. Show that for each integer k 2 1, if P(k) is true, then P(k + 1) is true: gt+1 - 3 , where f,, f, fa... is a sequence defined by the recurrence relation et k be any integer with k z 1, and suppose that P(k) is true. In other words, suppose that f,- f, -f + 2* for each integer k 2 2, with initial condition f, - 1. This is the inductive hypothesis.] Ne must show that P(k + 1) is true. In other words, we must show that f. 2=2 – 3 Now the left-hand side of P(k + 1) is f - f, +2* + 1 by definition of fi, fs, f., + 2* +1 by inductive hypothesis V - 3 by the laws of algebra and this is the right-hand side of P(k + 1). Hence the inductive step is complete. Thus, both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 56EQ