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# More Proof by induction(the rearranging bit)

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1. http://www.physicsandmathstutor.com/...%20Edexcel.pdf

question 8

i can't for the life of me see an end to get to the result

i tried messing about by taking "factors out and compasating by adding or taking away what was needed to make it equal and i've also tried expanding everything and trying to figure out what i need and taking out what i need which should leave me with the answer. But it hasn't. Expanding things is what i've done down below

*looks at what i have to prove

*sees he needs

*take out what he needs

so what can i do to get a 1? either i screwed up in my calculations or i'm doing it wrong(or the unrare case of both).
2. (Original post by huiop)
...
What are you doing on your first line...? That's wrong.

for
3. (Original post by huiop)
http://www.physicsandmathstutor.com/...%20Edexcel.pdf

question 8

i can't for the life of me see an end to get to the result

i tried messing about by taking "factors out and compasating by adding or taking away what was needed to make it equal and i've also tried expanding everything and trying to figure out what i need and taking out what i need which should leave me with the answer. But it hasn't. Expanding things is what i've done down below

*looks at what i have to prove

*sees he needs

*take out what he needs

so what can i do to get a 1? either i screwed up in my calculations or i'm doing it wrong(or the unrare case of both).
Your first line is wrong. Have another look at it.
4. (Original post by RDKGames)
What are you doing on your first line...? That's wrong.

for
(Original post by notnek)
Your first line is wrong. Have another look at it.
When u spend about 10 mins typing that out all to find that your first line was wrong ;____;

-.-' i subbed k into the oh well.

so that means if when n=k

then

I don't see where i can go with this though... i don't know what value of n U_k takes and i can't see a way to arrange what i have back into the U_n, what i have to prove.
5. (Original post by huiop)
When u spend about 10 mins typing that out all to find that your first line was wrong ;____;

-.-' i subbed k into the oh well.

so that means if when n=k

then

I don't see where i can go with this though... i don't know what value of n U_k takes and i can't see a way to arrange what i have back into the U_n, what i have to prove.
That's because you're trying to test something for k+1 when you're not required to!

First test the wanted result for
Then assume that result to be true for and write it out for which should be in terms of k.
Substitute that in for and rearrange for in the first equation.
You should get the required result to be in terms of k. Since we assumed that the result is true for , it has to be true in terms of n.

This is a slightly different proof than the summation of series. Been a while since I've proven these types by induction so I could be wrong but it makes sense to do it this way. I'm sure others will correct me in the case if I am.
6. (Original post by huiop)
When u spend about 10 mins typing that out all to find that your first line was wrong ;____;

-.-' i subbed k into the oh well.

so that means if when n=k

then

I don't see where i can go with this though... i don't know what value of n U_k takes and i can't see a way to arrange what i have back into the U_n, what i have to prove.
An induction proof will always involve you assuming the thing you're trying to prove for e.g. n = k.

If you're ever stuck with induction, check you've done the initial steps:

Have I proved the base case?
Have I assumed true for n = k ?
7. (Original post by RDKGames)
That's because you're trying to test something for k+1 when you're not required to!

First test the wanted result for
Then assume that result to be true for and write it out for which should be in terms of k.
Substitute that in for and rearrange for in the first equation.
You should get the required result to be in terms of k. Since we assumed that the result is true for , it has to be true in terms of n.

This is a slightly different proof than the summation of series. Been a while since I've proven these types by induction so I could be wrong but it makes sense to do it this way. I'm sure others will correct me in the case if I am.
so i don't need to test for k+1 because is already the next term?
but then what have i assumed is true for n=k?
(Original post by notnek)
An induction proof will always involve you assuming the thing you're trying to prove for e.g. n = k.

If you're ever stuck with induction, check you've done the initial steps:

Have I proved the base case?
Have I assumed true for n = k ?
Ah ok but what about n+1?
8. (Original post by huiop)
so i don't need to test for k+1 because is already the next term?
but then what have i assumed is true for n=k?
No, you are not proving the next term of a sequence; you are proving that (i.e. the nth term can be expressed in terms of n). If you were proving for the next term, then you would need to prove for

You are assuming that is true for and getting an expression for which you then use as a substitution. This is the 'induction' part.
9. (Original post by RDKGames)
No, you are not proving the next term of a sequence; you are proving that . If you were proving for the next term, then you would need to prove for

You are assuming that is true for and getting an expression for which you then use as a substitution. This is the 'induction' part.
ah i gathered this just as you wrote this post ^-^

Thanks!

Also this is probably a stupid question but what if i sub in n=k and it's not true? o.o
10. (Original post by huiop)
ah i gathered this just as you wrote this post ^-^

Thanks!

Also this is probably a stupid question but what if i sub in n=k and it's not true? o.o
What do you mean? You're not subbing in . You're assuming it's true for n=k and subbing in . If you don't get the result then you're doing it wrong; or if you're 100% sure you do it right and others agree with you, then the proof does not hold. But other proof methods would be used before this is established.
11. (Original post by RDKGames)
What do you mean? You're not subbing in . You're assuming it's true for n=k and subbing in . If you don't get the result then you're doing it wrong; or if you're 100% sure you do it right and others agree with you, then the proof does not hold. But other proof methods would be used before this is established.

i'm even more confused now.so far i've assumed true for n=k then

and

12. (Original post by huiop)

i'm even more confused now.so far i've assumed true for n=k then

and

Again, you're trying to do some proof for with the next term of the sequence. That's not what we're looking for. Assuming it's true for n=k:

and use this as a substitution into
13. (Original post by RDKGames)
Again, you're trying to do some proof for with the next term of the sequence. That's not what we're looking for. Assuming it's true for n=k:

and use this as a substitution into
so ??
14. (Original post by huiop)
so ??
Yes. (though it should be [(k+1)-1] in the bracket on the left)
15. (Original post by huiop)

i'm even more confused now.so far i've assumed true for n=k then

and

You should have which does in fact simplify and gives the required result.
16. (Original post by RDKGames)
Yes. (though it should be [(k+1)-1] in the bracket on the left)
....
(Original post by B_9710)
You should have which does in fact simplify and gives the required result.
oh i see so it does.... thanks!
17. (Original post by huiop)

oh i see so it does.... thanks!
Keep in mind that would give you the . For you'd need to map on the right side. This would give you the exact same thing as mine would and proof would hold.
18. (Original post by RDKGames)
Keep in mind that would give you the . For you'd need to map on the right side. Would give you the exact same thing as mine would and proof would hold.
The proof for showing the results for or are equivalent, they're exactly the same. With induction all you need to show is that if the result is true for then using logical arguments it implies that the result is also true for .
19. (Original post by B_9710)
The proof for showing the results for or are equivalent, they're exactly the same. With induction all you need to show is that if the result is true for then using logical arguments it implies that the result is also true for .
I understand that but I think the mark scheme would require you to have at the end rather than because it fits their answer that way. Not sure if they're picky about that, though.
20. (Original post by RDKGames)
I understand that but I think the mark scheme would require you to have at the end rather than because it fits their answer that way. Not sure if they're picky about that, though.
It's the same thing, for one of them and for the other .

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