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Please help me with apply the inductive hypothesis to fk and fk-1 and complete the proof.

Please help me with apply the inductive hypothesis to fk and fk-1 and complete the proof.

Advanced Engineering Mathematics
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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Please help me with apply the inductive hypothesis to fk and fk-1 and complete the proof.
Suppose that fo, f. f is a sequence defined as follows, fo - 5, f,- 16, f - 7f-- 10f-, for every integer k 22 Prove that f- 3. 2" + 2. 5" for each integer n 2 0. Proof by strong mathematical induction: Let the property P(n) be the equation f- 3. 2" + 2.5". We will show that P(n) is true for every integer n 2 0. Show that P(0) and P(1) are true: Select P(0) from the choices below. o P(0) - 3- 2° + 2. 5° • f, -3- 2° + 2 -50 O fo-5 P(O) - fo Select P(1) from the choices below. P(1) = f, -16 f, -3-24 + 2-5? O P(1) - 3- 21 + 2 - 51 P(0) and P(1) are true because 3- 2° + 2. 5° - 5 and 3- 21 + 2· 5 - 16. Show that for every integer k 2 1, if P() is true for each integer i from 0 through k, then P(k + 1) is true: Let k be any integer with k 2 1, and suppose that for every integer / with 0 sisk, f, -3.2' + 2-5 This is the (inductive hypothesis v We must show that f1 -2*+1+2-5k+1 Now, by definition of fo f, f -. fk+1= -10f1 Apply the inductive hypothesis to f and f- 1 and complete the proof as a free response. (Submit a file with a maximum size of 1 MB.) Choose File No file chosen
Transcribed Image Text:Suppose that fo, f. f is a sequence defined as follows, fo - 5, f,- 16, f - 7f-- 10f-, for every integer k 22 Prove that f- 3. 2" + 2. 5" for each integer n 2 0. Proof by strong mathematical induction: Let the property P(n) be the equation f- 3. 2" + 2.5". We will show that P(n) is true for every integer n 2 0. Show that P(0) and P(1) are true: Select P(0) from the choices below. o P(0) - 3- 2° + 2. 5° • f, -3- 2° + 2 -50 O fo-5 P(O) - fo Select P(1) from the choices below. P(1) = f, -16 f, -3-24 + 2-5? O P(1) - 3- 21 + 2 - 51 P(0) and P(1) are true because 3- 2° + 2. 5° - 5 and 3- 21 + 2· 5 - 16. Show that for every integer k 2 1, if P() is true for each integer i from 0 through k, then P(k + 1) is true: Let k be any integer with k 2 1, and suppose that for every integer / with 0 sisk, f, -3.2' + 2-5 This is the (inductive hypothesis v We must show that f1 -2*+1+2-5k+1 Now, by definition of fo f, f -. fk+1= -10f1 Apply the inductive hypothesis to f and f- 1 and complete the proof as a free response. (Submit a file with a maximum size of 1 MB.) Choose File No file chosen
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