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(d) Consider the sequences (In), (n) defined recursively as follows: In+1 = In 2yn Yn+1 = Yn + 2x for n ≥1, ₁ = 1, 12. i. Calculate 12, y2 and 13, Y3- ii. Show - using induction or otherwise - that for any natural number n, In+Yni = (1+2i)". iii. Hence or otherwise, show that for any natural n, Zn = (V5)” cos(narctan2), n = (V5)” sin(n arctan2).

(d) Consider the sequences (In), (n) defined recursively as follows: In+1 = In 2yn Yn+1 = Yn + 2x for n ≥1, ₁ = 1, 12. i. Calculate 12, y2 and 13, Y3- ii. Show - using induction or otherwise - that for any natural number n, In+Yni = (1+2i)". iii. Hence or otherwise, show that for any natural n, Zn = (V5)” cos(narctan2), n = (V5)” sin(n arctan2).

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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(d) Consider the sequences (In), (n) defined recursively as follows: In+1 = In 2yn Yn+1 = Yn + 2x for n ≥1, ₁ = 1, 12. i. Calculate 12, y2 and 13, Y3- ii. Show - using induction or otherwise - that for any natural number n, In+Yni = (1+2i)". iii. Hence or otherwise, show that for any natural n, Zn = (V5)” cos(narctan2), n = (V5)” sin(n arctan2).
Transcribed Image Text:(d) Consider the sequences (In), (n) defined recursively as follows: In+1 = In 2yn Yn+1 = Yn + 2x for n ≥1, ₁ = 1, 12. i. Calculate 12, y2 and 13, Y3- ii. Show - using induction or otherwise - that for any natural number n, In+Yni = (1+2i)". iii. Hence or otherwise, show that for any natural n, Zn = (V5)” cos(narctan2), n = (V5)” sin(n arctan2).
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