Let (ao, a₁,..., an) be a finite simple continued fraction, and let pn and In be the numbers defined in Exercise 10. Prove that Pn9n-1-Pn-19n = (-1)"-1 and for n= 1,..., N. Prove that if a; E Z for i = 0, 1,..., N, then (Pn) 9n) 1 for n = 0, 1,..., N. =
Let (ao, a₁,..., an) be a finite simple continued fraction, and let pn and In be the numbers defined in Exercise 10. Prove that Pn9n-1-Pn-19n = (-1)"-1 and for n= 1,..., N. Prove that if a; E Z for i = 0, 1,..., N, then (Pn) 9n) 1 for n = 0, 1,..., N. =
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:10. Let (ao, a₁,.., aN) be a finite simple continued fraction. Define
po = ao,
P1 = a1a0 + 1,
and
Define
and
Prove that
Pn anPn-1+Pn-2
90 = 1,
91 = a1,
for
n = 2,..., N.
1.3 The Euclidean Algorithm and Continued Fractions 23
qn anqn-1 +9n-2 for n = 2,..., N.
Pn
qn
(ao, a₁,..., an) =
for n =
= 0, 1,..., N. The continued fraction (ao, a₁,..., an) is called
the nth convergent of the continued fraction (ao, a1,..., an).
Transcribed Image Text:12. Let (ao, a1,..., an) be a finite simple continued fraction, and let pn
and qn be the numbers defined in Exercise 10. Prove that
Pn9n-1-Pn-19n = (-1)"-1
and for n = 1,..., N. Prove that if a; € Z for i = 0, 1,..., N, then
(Pn, n) = 1 for n = 0, 1,..., N.
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