Induction working out... (1 Viewer)

[HeroicPandas]

Hi, can someone please check my WORKING OUT?



This is what I did:

Step 1: Show true for n=3,





Hence true for n=3

Step 2: Assume true for n=k, (anything else I need to write?)

etc, etc

The question is, when do I need to sub in 2 or more values of 'n' for STEP 1?

For the STEP 2, the assumption, do I need to specify values of 'k'? The recurrence relation has values of 'n' that does not satisfy the thing im proving - do I just say 'assume true for all k graeter or equal to 1'?

Thanks in advance

 

[Carrotsticks]

For your base cases, you could have used T_1 and T_2.

You need 2 base cases because you will be using two terms, each of which we are 'performing induction on'.

To do the question, you need 2 inductive hypothesis, n=k-1 and n=k, and use BOTH of these to prove n=k+1. Because you have 2 inductive hypotheses, you will need 2 base cases. One for each hypothesis.

Your recurrence relation is of second order, so you will need 2 base cases.

 

[HeroicPandas]

For your base cases, you could have used T_1 and T_2.

You need 2 base cases because you will be using two terms, each of which we are 'performing induction on'.

To do the question, you need 2 inductive hypothesis, n=k-1 and n=k, and use BOTH of these to prove n=k+1. Because you have 2 inductive hypotheses, you will need 2 base cases. One for each hypothesis.

Your recurrence relation is of second order, so you will need 2 base cases.

Ah i see..
Will I lose marks if I dont clearly write out: assume true for n=k-1? (for step 2)

Can you assume n=k-1 when doing the n=k+1 step without saying that you assumed it?

 

[Carrotsticks]

Ah i see..
Will I lose marks if I dont clearly write out: assume true for n=k-1? (for step 2)

Can you assume n=k-1 when doing the n=k+1 step without saying that you assumed it?

If you're unlucky then yes because one of the core aspects of strong induction is the fact that you need more than 1 induction hypothesis.

 

[HeroicPandas]

If you're unlucky then yes because one of the core aspects of strong induction is the fact that you need more than 1 induction hypothesis.

Ok thanks!

Do I need to discuss the values of 'k' for my assumption?

 

[Carrotsticks]

Ok thanks!

Do I need to discuss the values of 'k' for my assumption?

What do you mean by discussing them?

 

[HeroicPandas]

What do you mean by discussing them?

Do I need to mention values of 'k'? Or just forget about all that?

Because in the question, the recurrence relation is for n ≥3 and the thing im trying to prove is n≥1

 

[Carrotsticks]

Do I need to mention values of 'k'? Or just forget about all that?

Because in the question, the recurrence relation is for n ≥3 and the thing im trying to prove is n≥1

Not essential.

Good to have, but not essential.

 

[HeroicPandas]

Not essential.

Good to have, but not essential.

Ok very nice! Thanks for the help! Cleared up all the fog

 

[Carrotsticks]

Ok very nice! Thanks for the help! Cleared up all the fog

Just to make sure that you've gotten it, can you now type up a full answer so I can see whether it's all good or not?

 

[HeroicPandas]

Just to make sure that you've gotten it, can you now type up a full answer so I can see whether it's all good or not?

ok

 

[HeroicPandas]

Show true for n=1,

Show true for n=2,

Hence true for n=1 and n=2

Assume true for n=k-1 and n=k
i.e.



Prove true for n=k+1,
i.e.




Hence true by mathematical induction for all integers greater or equal to 1

 

[Carrotsticks]

Looks good =)

Except ditch the very last line, that is unnecessary in the HSC. The markers don't even read the last line (conclusion).

 

[HeroicPandas]

Looks good =)

Except ditch the very last line, that is unnecessary in the HSC. The markers don't even read the last line (conclusion).

ok cool