This question is about the sequence {an} given by An =2n^2 −11n−6, n=1,2,....
Throughout the question you may use without proof any results from the notes that you need.
(a) Find A1, A2, A3 and A4.
(b) Give examples to show that {An} is neither increasing nor decreasing.
(c) Show that {An} is eventually monotonic and state whether this is monotonic increasing or monotonic decreasing.
Any Help will be appreciated!
Throughout the question you may use without proof any results from the notes that you need.
(a) Find A1, A2, A3 and A4.
(b) Give examples to show that {An} is neither increasing nor decreasing.
(c) Show that {An} is eventually monotonic and state whether this is monotonic increasing or monotonic decreasing.
Any Help will be appreciated!
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This question is about the sequence {an} given by An =2n^2 −11n−6, n=1,2,....
Throughout the question you may use without proof any results from the notes that you need.
(a) Find A1, A2, A3 and A4.
a1=2-11-6=-15
a2=8-22-6=-20
a3=18-33-6=-21
a4=32-44-6=-18
(b) Give examples to show that {An} is neither increasing nor decreasing.
if a(n) is increasing then a2>a1. However, -20>-15 obviously false
if a(n) is decreasing, a4
(c) Show that {An} is eventually monotonic and state whether this is monotonic increasing or monotonic decreasing.
a(n)=2n^2-11n-6
a(n+1)=2(n+1)^2-11(n+1)-6
=2(n^2+2n+1)-11n-11-6
=2n^2+4n+2-11n-17
since 2n^2>0, a(n) will eventually monotonically increase
Throughout the question you may use without proof any results from the notes that you need.
(a) Find A1, A2, A3 and A4.
a1=2-11-6=-15
a2=8-22-6=-20
a3=18-33-6=-21
a4=32-44-6=-18
(b) Give examples to show that {An} is neither increasing nor decreasing.
if a(n) is increasing then a2>a1. However, -20>-15 obviously false
if a(n) is decreasing, a4
a(n)=2n^2-11n-6
a(n+1)=2(n+1)^2-11(n+1)-6
=2(n^2+2n+1)-11n-11-6
=2n^2+4n+2-11n-17
since 2n^2>0, a(n) will eventually monotonically increase
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(a) just sub in the number
A1 = 2[1]^2 − 11[1] − 6 = 2 - 11 - 6 = -15
A2 = 2[2]^2 − 11[2] − 6] = 8 - 22 - 6 = -20
A3 = 2[3]^2 − 11[3] − 6 = 18 - 33 - 6 = -21
A4 = 2[4]^2 − 11[4] − 6 = 32 - 44 - 6 = -18
(b) this is the kind of crap question that math teachers love but students hate, and it really doesn't teach much either.
All they want you to realize here is that the sequence both increases and decreases at different points. Since it does both, technically is is not an increasing function OR a decreasing function.
An increasing function can never decrease at any instant.
This can be easily observed as the change from A2 to A3 is -1, but the change from A3 ro A4 is +3.
So on the interval from A3-A4, the sequence increases in value, while for the others, it decreases.
3
I don't recall ever hearing about monotonic functions before. SO that one I leave to you, or someone else.
All the other answer does is basically show that there is an inductive process at work here.
A1 = 2[1]^2 − 11[1] − 6 = 2 - 11 - 6 = -15
A2 = 2[2]^2 − 11[2] − 6] = 8 - 22 - 6 = -20
A3 = 2[3]^2 − 11[3] − 6 = 18 - 33 - 6 = -21
A4 = 2[4]^2 − 11[4] − 6 = 32 - 44 - 6 = -18
(b) this is the kind of crap question that math teachers love but students hate, and it really doesn't teach much either.
All they want you to realize here is that the sequence both increases and decreases at different points. Since it does both, technically is is not an increasing function OR a decreasing function.
An increasing function can never decrease at any instant.
This can be easily observed as the change from A2 to A3 is -1, but the change from A3 ro A4 is +3.
So on the interval from A3-A4, the sequence increases in value, while for the others, it decreases.
3
I don't recall ever hearing about monotonic functions before. SO that one I leave to you, or someone else.
All the other answer does is basically show that there is an inductive process at work here.