Suppose that for f₁ f2. is a sequence defined as follows. fo= 5, f₁ = 16, fk = 7fk-1-10fk - 2 for every integer k ≥ 2 Prove that f= 3.2" +2.5" for each integer n ≥ 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 ≥ 0. Show that P(0) and P(1) are true: Select P(0) from the choices below. OP(0) = 3.20 +2.5⁰ ○ P(0) = fo Of=5 Of=3.20 +2.5⁰ Select P(1) from the choices below. O P(1) = f₁ OP(1) = 3.2¹ +2.5¹ Of₁ = 3.2¹+2.5¹ O f₁ = 16 P(0) and P(1) are true because 3-20 +2.505 and 3.2¹ +2.5¹ = 16. Show that for every integer k ≥ 1, if P(i) is true for each integer i from 0 through k, then P(k + 1) is true: Let k be any integer with k ≥ 1, and suppose that for every integer i with 0 ≤ i ≤k, f, = . This is the ---Select--- Now, by definition of for f₁ f₂. fk+1= Apply the inductive hypothesis to f and fk - 1 and complete the proof as a free response. (Submit a file with a maximum size of 1 MB.) v We must show that fk + 1 =
Suppose that for f₁ f2. is a sequence defined as follows. fo= 5, f₁ = 16, fk = 7fk-1-10fk - 2 for every integer k ≥ 2 Prove that f= 3.2" +2.5" for each integer n ≥ 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 ≥ 0. Show that P(0) and P(1) are true: Select P(0) from the choices below. OP(0) = 3.20 +2.5⁰ ○ P(0) = fo Of=5 Of=3.20 +2.5⁰ Select P(1) from the choices below. O P(1) = f₁ OP(1) = 3.2¹ +2.5¹ Of₁ = 3.2¹+2.5¹ O f₁ = 16 P(0) and P(1) are true because 3-20 +2.505 and 3.2¹ +2.5¹ = 16. Show that for every integer k ≥ 1, if P(i) is true for each integer i from 0 through k, then P(k + 1) is true: Let k be any integer with k ≥ 1, and suppose that for every integer i with 0 ≤ i ≤k, f, = . This is the ---Select--- Now, by definition of for f₁ f₂. fk+1= Apply the inductive hypothesis to f and fk - 1 and complete the proof as a free response. (Submit a file with a maximum size of 1 MB.) v We must show that fk + 1 =
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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Transcribed Image Text:Suppose that for f₁, f₂
fo
=
is a sequence defined as follows.
= 16
5, f₁ = 16,
= 7fk - 1
10fk
Prove that f= 3 - 2ª + 2 · 5″ for each integer n ≥ 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 ≥ 0.
Show that P(0) and P(1) are true:
Select P(0) from the choices below.
O P(0) = 3.20 +2.5⁰
O P(0) = fo
To = 5
O f = 3.20 +2.5⁰
- 2
Select P(1) from the choices below.
O P(1) = f₁
OP(1) = 32¹ + 2 · 5¹
f₁ = 3.2¹ +2 5¹
f₁
P(0) and P(1) are true because 3 20 +2.50 = 5 and 3 · 2¹ + 2.5¹ = 16.
Show that for every integer k ≥ 1, if P(i) is true for each integer i from 0 through k, then P(k+ 1) is true:
for every integer k ≥ 2
Now, by definition of fo, f₁, f₂,
fk+ 1 =
Let k be any integer with k ≥ 1, and suppose that for every integer i with 0 ≤ i ≤ k, f; =
Apply the inductive hypothesis to fk and fk - 1
This is the --Select---
and complete the proof as a free response. (Submit a file with a maximum size of 1 MB.)
We must show that f
+1
=
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