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Theorem 2. Suppose that {xn} is solutions of (3 ) and the initial value xo is an arbitrary nonzero real number. Let xo = a .Then ,by using the notations(III), the solutions of (3) are given by: (n- a A1 n-1 a Xn = An i =1 II 4-2 (IV) where xo = a and n > 2 Proof. Firstly, a (х о)9 -1 + a A1 Now ,by mathematical induction , we will prove that equations{IV} are true for n > 2. In the begining we try to prove that equations{IV} are true for n=2. C- 2-1 а a q-2 2 = (x 1)4-1 + a -1 + a A1A2 A2 i =1 Now suppose that the equations{IV} is true for n = r.This means that (-- i r-1 a Xr = A, i =1 II 49 Finally we prove that the equations{IV} is true for n =r+ 1. Xr Xr+1 = +a

Theorem 2. Suppose that {xn} is solutions of (3 ) and the initial value xo is an arbitrary nonzero real number. Let xo = a .Then ,by using the notations(III), the solutions of (3) are given by: (n- a A1 n-1 a Xn = An i =1 II 4-2 (IV) where xo = a and n > 2 Proof. Firstly, a (х о)9 -1 + a A1 Now ,by mathematical induction , we will prove that equations{IV} are true for n > 2. In the begining we try to prove that equations{IV} are true for n=2. C- 2-1 а a q-2 2 = (x 1)4-1 + a -1 + a A1A2 A2 i =1 Now suppose that the equations{IV} is true for n = r.This means that (-- i r-1 a Xr = A, i =1 II 49 Finally we prove that the equations{IV} is true for n =r+ 1. Xr Xr+1 = +a

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.3: Quadratic Equations
Problem 81E
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A more general form of difference equation (2) can take the following nonlinear rational difference equations : Xn+1 (xn)- +a (3) where (xo)9-1 Now consider the following notations + -a. A1 = a9-1 + a A2 = a"-1 + a Af- q-1 (p-2 (q-1)(q-2) Ap = = a9-1 + a 'p-1 (III) i=1 where p > 3.
Transcribed Image Text:A more general form of difference equation (2) can take the following nonlinear rational difference equations : Xn+1 (xn)- +a (3) where (xo)9-1 Now consider the following notations + -a. A1 = a9-1 + a A2 = a"-1 + a Af- q-1 (p-2 (q-1)(q-2) Ap = = a9-1 + a 'p-1 (III) i=1 where p > 3.
Theorem 2. Suppose that {xn} is solutions of (3 ) and the initial value xo is an arbitrary nonzero real number . Let xo = a .Then ,by using the notations(III) , the solutions of (3) are given by: a A1 n-1 a -2 Xn (IV) An i =1 where xo = a and n > 2 Proof. Firstly, a x 1 = (x 0)ª-1 + a A1 Now ,by mathematical induction , we will prove that equations{IV} are true for n > 2. In the begining we try to prove that equations{IV} are true for n=2 . x 1 2-1 a a A II 42 x 2 (x 1)ª-1 + a -1 + a A1 A2 A2 i =1 Now suppose that the equations{IV} is true for n = r.This means that r-1 a q-2 Xr = A, i =1 Finally we prove that the equations{IV} is true for n = r + 1. Xr+1 = -1 +a
Transcribed Image Text:Theorem 2. Suppose that {xn} is solutions of (3 ) and the initial value xo is an arbitrary nonzero real number . Let xo = a .Then ,by using the notations(III) , the solutions of (3) are given by: a A1 n-1 a -2 Xn (IV) An i =1 where xo = a and n > 2 Proof. Firstly, a x 1 = (x 0)ª-1 + a A1 Now ,by mathematical induction , we will prove that equations{IV} are true for n > 2. In the begining we try to prove that equations{IV} are true for n=2 . x 1 2-1 a a A II 42 x 2 (x 1)ª-1 + a -1 + a A1 A2 A2 i =1 Now suppose that the equations{IV} is true for n = r.This means that r-1 a q-2 Xr = A, i =1 Finally we prove that the equations{IV} is true for n = r + 1. Xr+1 = -1 +a
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