Lhe blahks Ih the föllowing proof, which shows that the sequence defined by the recurrence relation k = fk-1+ 2* for each integer k 22 atisfies the following formula. f. = 2" +1- 3 for every integer n21 "roof (by mathematical induction): uppose f, f, f... is a sequence that satisfies the recurrence relation f = f-1+ 2* for each integer k2 2, with initial condition f, = 1. Ve need to show that when the sequence f,, fa, fa,... is defined in this recursive way, all the terms in the seguence 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 n 2 1. Show that P(1) is true: The left-hand side of P(1) is 4-3 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) Show that for each integerkz 1, if P(k) is true, then P(k + 1) is true: Let k be any integer with k2 1, and suppose that P(k) is true. In other words, suppose that f, =ok+1_3 where f, f, f... is a sequence defined by the recurre f, = f, + 2* for each integer k 2 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 f ,= k+2 3 Now the left-hand side of P(k + 1) is k+1 = + 2*+ 1 k+1-3 by definition of fs, fa, fs, .. + 2k +1 by inductive hypothesis %3D ok+1 = 2 ak+2 3 by the laws of algebra and this is the right-hand side of P(k + 1). Hence the inductive step is complete.
Lhe blahks Ih the föllowing proof, which shows that the sequence defined by the recurrence relation k = fk-1+ 2* for each integer k 22 atisfies the following formula. f. = 2" +1- 3 for every integer n21 "roof (by mathematical induction): uppose f, f, f... is a sequence that satisfies the recurrence relation f = f-1+ 2* for each integer k2 2, with initial condition f, = 1. Ve need to show that when the sequence f,, fa, fa,... is defined in this recursive way, all the terms in the seguence 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 n 2 1. Show that P(1) is true: The left-hand side of P(1) is 4-3 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) Show that for each integerkz 1, if P(k) is true, then P(k + 1) is true: Let k be any integer with k2 1, and suppose that P(k) is true. In other words, suppose that f, =ok+1_3 where f, f, f... is a sequence defined by the recurre f, = f, + 2* for each integer k 2 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 f ,= k+2 3 Now the left-hand side of P(k + 1) is k+1 = + 2*+ 1 k+1-3 by definition of fs, fa, fs, .. + 2k +1 by inductive hypothesis %3D ok+1 = 2 ak+2 3 by the laws of algebra and this is the right-hand side of P(k + 1). Hence the inductive step is complete.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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