MX2 Mathematical Induction (2 Viewers)

[blaze.bluetane]

Has anyone got any difficult/challenging MX2 level mathematical induction questions? Maybe from a trial or textbook or whatever. Would be much appreciated.

 

[abc123doremi]

there are many 4U geometical induction questions in the extension section of cambridge 3U. Chapter 8 i think it was

or not. not really sure but probably

 

[blaze.bluetane]

there are many 4U geometical induction questions in the extension section of cambridge 3U. Chapter 8 i think it was

or not. not really sure but probably

I have that textbook I will check it out thanks.

 

[Affinity]

prove that a cube can be divided into exactly n cubes for each n >= 100, (the sizes of the cubes need not be the same)

 

[lychnobity]

Has anyone got any difficult/challenging MX2 level mathematical induction questions? Maybe from a trial or textbook or whatever. Would be much appreciated.

1) Suppose x>0, y>0

Rtp: 4/s ≤ 1/x + 1/y, where s = x + y

2) Suppose that x_i > 0 for i = 1,2...n, where n≥2

Prove by induction that:
1/x₁ + 1/x₂ + ... + 1/x_n ≥ n²/s

where s = x₁ + x₂ + ... + x_n

3) Prove by MI that:
a) 1 + 1/2² + 1/3² + ... + 1/n² ≤ 2 - 1/n for all n≥1

b) 1 + 1/2² + 1/3² + ... + 1/n² > 3/2 - 1/(n+1) for all n≥1

c) Not really necessary, but came with the question
Show that 1.49 < 1 + 1/2² + 1/3² + ... + 1/100² < 1.99

4) Prove by MI for all integers n≥1 that

(a₁ + a₂ + ... + a_n)/2 ≥ (a₁.a₂...a₂)^1/n

Where a₁, a₂, ..., a_n > 0

Most of these questions are in Fundamental Mathematics - Terry Lee, with solutions. Actually, the later questions are quite hard, might want to try some.

 

[Affinity]

For n = 1,2,3,4 find the number of ways of placing n letters in n envelops such that no letter is in the correct envelope ( assume the letters and envelopes form pairs)

Prove that in general for n >= 0 the following formula gives the number of ways

 

[addikaye03]

For n = 1,2,3,4 find the number of ways of placing n letters in n envelops such that no letter is in the correct envelope ( assume the letters and envelopes form pairs)

Prove that in general for n >= 0 the following formula gives the number of ways

i know if theres (n+1) envelopes that need to be placed in n letterboxes (ie. more envelopes than letterboxes) you can use a theory called "the pigeonhole theory"

 

[LordPc]

its actually called the pigeonhole principle, since 'theory' means it has no proof. like string theory

 

[addikaye03]

its actually called the pigeonhole principle, since 'theory' means it has no proof. like string theory

ahh yeah, my bad

 

[blaze.bluetane]

4) Prove by MI for all integers n≥1 that

(a₁ + a₂ + ... + a_n)/2 ≥ (a₁.a₂...a₂)^1/n

Where a₁, a₂, ..., a_n > 0

should this be...
(a₁ + a₂ + ... + a_n)/n ≥ (a₁.a₂...a_n)^1/n

Arithmetic mean is always greater than or equal to a geometric mean?


By the way thank you to everyone. Anyone else with good induction questions please do not hesitate to put them up.

Also... solutions? [I think I might invest in Terry Lee's book(s) - they are the only ones I don't have... they are good right?]

 

[gurmies]

should this be...
(a₁ + a₂ + ... + a_n)/n ≥ (a₁.a₂...a_n)^1/n

Arithmetic mean is always greater than or equal to a geometric mean?


By the way thank you to everyone. Anyone else with good induction questions please do not hesitate to put them up.

Also... solutions? [I think I might invest in Terry Lee's book(s) - they are the only ones I don't have... they are good right?]

Yep, should be /n.

 

[lychnobity]

Yep, should be /n.

Yeah, sorry, my bad.

 

[lychnobity]

Also... solutions? [I think I might invest in Terry Lee's book(s) - they are the only ones I don't have... they are good right?]

Yeah, all questions in his book have solutions (may be a bit confusing at first, but at least they're there). I think they're good books, caters more to harder questions (btw the questions I posted were in the challenge problems section)

 

[cutemouse]

[I think I might invest in Terry Lee's book(s) - they are the only ones I don't have... they are good right?]

I don't think he sells it anymore So gl getting it 2nd hand.

 

[lychnobity]

I don't think he sells it anymore So gl getting it 2nd hand.

He recently had a new batch come in. If any of you guys need copies, I can buy it on your behalf, and send it to you (I am tutored by him).

 

[cutemouse]

(I am tutored by him).

gg then. I read here that he perves on females students.

 

[lychnobity]

gg then. I read here that he perves on females students.

By 'perving' though, it's more a hit on the head, pinch on the ear etc. And the leaning, it's a height issue, he's only about 145cm, so when he leans over to check your work, it comes across as invading personal space.

/shrug

____________________________

This is question 8 from the Moriah College 2001 trial

 

[Affinity]

i know if theres (n+1) envelopes that need to be placed in n letterboxes (ie. more envelopes than letterboxes) you can use a theory called "the pigeonhole theory"

this has nothing to do with the question

 

[blaze.bluetane]

You seem to be able to buy Terry Lee's books through paypal on
HSC Coaching
Isn't that his site?
What's the hitch?

 

[lychnobity]

You seem to be able to buy Terry Lee's books through paypal on
HSC Coaching
Isn't that his site?
What's the hitch?

Yeah that's his site. A few months ago, they were out of print so the textbook based on the new syllabus would come out.

But the new syllabus got scrapped, and he reprinted the old book 3 weeks ago.

So, no hitch.