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(2) Let an be the sequence defined by: a₁ = 1 a2 = 2 an an-1 + 2an-2 for n ≥ 3. (a) Find a3,..., as. (b) Use induction to prove that an> 0 for all n € N. an < 1 for all n EN. (c) Use induction to prove that an+1 (d) Use (b) and (c) to conclude that an

(2) Let an be the sequence defined by: a₁ = 1 a2 = 2 an an-1 + 2an-2 for n ≥ 3. (a) Find a3,..., as. (b) Use induction to prove that an> 0 for all n € N. an < 1 for all n EN. (c) Use induction to prove that an+1 (d) Use (b) and (c) to conclude that an

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 53E: Given the recursively defined sequence a1=0,a2=30, and an=8an115an2, use complete induction to prove...
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(2) Let a(n) be the sequence defined by:

a1 = 1

a2 = 2

an = an−1 + 2an−2 for n ≥3.

(a) Find a3, . . . , a8.

(b) Use induction to prove that an > 0 for all n ∈N.

(c) Use induction to prove that an/an+1 < 1 for all n ∈N.

(d) Use (b) and (c) to conclude that an < an+1 for all n ∈N.

(2) Let an be the sequence defined by: a1 = 1 a2 = 2 an an-1 + 2an-2 for n > 3. (a) Find a3,.. (b) Use induction to prove that an > 0 for all n € N. (c) Use induction to prove that < 1 for all nЄN. .... ag. an an+1 (d) Use (b) and (c) to conclude that an < an+1 for all n € N.
Transcribed Image Text:(2) Let an be the sequence defined by: a1 = 1 a2 = 2 an an-1 + 2an-2 for n > 3. (a) Find a3,.. (b) Use induction to prove that an > 0 for all n € N. (c) Use induction to prove that < 1 for all nЄN. .... ag. an an+1 (d) Use (b) and (c) to conclude that an < an+1 for all n € N.
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