a) Let the sequence {an}n=1 is defined by a, = 4 and an+1 =5for n E N6-ani. Prove that {an}n=1 is a decresing sequence.ii. Prove that 0 < an for all n E N.iii. Find lim an.n-00iv. Describe the behaviour of the sequence {a,}=1, if a, = 1.b) For each of the following series, determine whether it converges or diverges.justifications and state the name of the convergence test you use.n2"En=1i.n!n2+2ii.n%=D12n2-п+5.

(a) The given sequence is an+1=56-an ∀ n∈ℕ and a1=4. To find: the answer of the given options.…

5 a) Let the sequence {an}=1 is defined by a₁ = 4 and an+1 =; for n E N 6-an i. Prove that {an}=1 is a decresing sequence. ii. Prove that 0 < an for all n E N. iii. Find lim an. 12-00 iv. Describe the behaviour of the sequence {an}n=1, if a₁ = 1.

5 a) Let the sequence {an}=1 is defined by a₁ = 4 and an+1 =; for n E N 6-an i. Prove that {an}=1 is a decresing sequence. ii. Prove that 0 < an for all n E N. iii. Find lim an. 12-00 iv. Describe the behaviour of the sequence {an}n=1, if a₁ = 1.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.2: Mathematical Induction

Problem 55E: The Fibonacci sequence fn=1,1,2,3,5,8,13,21,... is defined recursively by f1=1,f2=1,fn+2=fn+1+fn for...

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Question

a) Let the sequence {an}n=1 is defined by a, = 4 and an+1 =
5
for n E N
6-an
i. Prove that {an}n=1 is a decresing sequence.
ii. Prove that 0 < an for all n E N.
iii. Find lim an.
n-00
iv. Describe the behaviour of the sequence {a,}=1, if a, = 1.
b) For each of the following series, determine whether it converges or diverges.
justifications and state the name of the convergence test you use.
n2"
En=1
i.
n!
n2+2
ii.
n%=D1
2n2-п+5.

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