Suppose we have a recursive sequence ƒ1, ƒ2, ƒ3, . . .. For the purposes of this problem, it does not matter exactly how the fi are defined, only that they are recursively defined. = i=1 For integer n ≥ 1, let P(n) be the predicate that Σï-1 fi = 2fn+2 − 3. Don't worry about whether this predicate "makes sense"; we haven't defined the fi so you won't be able to "make sense" of the P(n). It's not important for this problem. Consider a proof by induction that Vn ≥ 1: P(n). Suppose we've gotten to the inductive step, and suppose that the first steps of the inductive step are k+1 k+1 Σ1 fi = f1 + f2 + Σh fi = f₁ + f2 + Σkt k+1, i=3 (fi−1 + fi−2) = f₁ + f2 + ¹ fi- 1 + k fi 2 f1 k+1 k+1 k = f1 + f2 + Σh_2 fi + [h=1² fi i=2 = f2 + Σ²²₁ fi+fi i=1 i=1 True or false: Based on the information given, we will need strong induction for this proof. True False
Suppose we have a recursive sequence ƒ1, ƒ2, ƒ3, . . .. For the purposes of this problem, it does not matter exactly how the fi are defined, only that they are recursively defined. = i=1 For integer n ≥ 1, let P(n) be the predicate that Σï-1 fi = 2fn+2 − 3. Don't worry about whether this predicate "makes sense"; we haven't defined the fi so you won't be able to "make sense" of the P(n). It's not important for this problem. Consider a proof by induction that Vn ≥ 1: P(n). Suppose we've gotten to the inductive step, and suppose that the first steps of the inductive step are k+1 k+1 Σ1 fi = f1 + f2 + Σh fi = f₁ + f2 + Σkt k+1, i=3 (fi−1 + fi−2) = f₁ + f2 + ¹ fi- 1 + k fi 2 f1 k+1 k+1 k = f1 + f2 + Σh_2 fi + [h=1² fi i=2 = f2 + Σ²²₁ fi+fi i=1 i=1 True or false: Based on the information given, we will need strong induction for this proof. True False
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 we have a recursive sequence ƒ1, ƒ2, ƒ3, . . ..
For the purposes of this problem, it does not matter exactly how the fi are defined,
only that they are recursively defined.
=
i=1
For integer n ≥ 1, let P(n) be the predicate that Σï-1 fi = 2fn+2 − 3.
Don't worry about whether this predicate "makes sense"; we haven't defined the fi
so you won't be able to "make sense" of the P(n). It's not important for this
problem.
Consider a proof by induction that Vn ≥ 1: P(n).
Suppose we've gotten to the inductive step, and suppose that the first steps of the
inductive step are
k+1
k+1
Σ1 fi = f1 + f2 + Σh fi
= f₁ + f2 + Σkt
k+1,
i=3
(fi−1 + fi−2)
= f₁ + f2 + ¹ fi- 1 + k fi 2
f1
k+1
k+1
k
= f1 + f2 + Σh_2 fi + [h=1² fi
i=2
= f2 + Σ²²₁ fi+fi
i=1
i=1
True or false: Based on the information given, we will need strong induction for this
proof.
True
False
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