The Sherminator
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#1
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I don't know how to proceed after a few steps in the question, and having had a look at the MS, I don't get it.

Its says: prove that  2^n > 2n  for    n >= 3

I said for n = k  2^k > 2k
n = k + 1  2^{k+1}  > 2k * 2
 2^{k+1} > 2k + 2k

But I don't know what to do next.

Help appreciated
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The Sherminator
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(Original post by Angus-Higgins)
Consider what you need to show, namely that:

 2^{k + 1} > 2(k + 1) .

Angus Higgins
I know that, but how do I proceed from the step I mentioned?
We can expand to say that  2^{k + 1} > 2k + 2

But there is a step from where I am stuck on, to that, and I don't understand it.
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TheRandomer
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1 is always less than k..

Idk what the mark scheme wants from you, really.. what does it say?
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DoMakeSayThink
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(Original post by The Sherminator)
I know that, but how do I proceed from the step I mentioned?
We can expand to say that  2^{k + 1} > 2k + 2

But there is a step from where I am stuck on, to that, and I don't understand it.
Well, you've already established 2^{k+1} > 2(k + k) As it only wants proof for n >= 3, you can safely assume that  2(k+k) > 2(k+1).
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The Sherminator
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4k is bigger than (2k + 2) so we change 2k + 2k to (2k + 2) and continue the proof? Why do we swap though, thats what is confusing me (sorry for sounding daft )


That is what the MS does as well.
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The Sherminator
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(Original post by DoMakeSayThink)
Well, you've already established 2^{k+1} > 2(k + k) As it only wants proof for n >= 3, you can safely assume that  2(k+k) > 2(k+1).
Thats it? We know that because k > 3, we can rule out the previous values?
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DoMakeSayThink
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(Original post by The Sherminator)
Thats it? We know that because k > 3, we can rule out the previous values?
Not quite sure what you mean by rule out the previous values, but because we know k > 3 > 1, we can just drop a 1 in, in place of the k, and it will strengthen the inequality. The way I originally phrased it might be a little more formal, but as long as you explain what you're doing, you're fine.
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The Sherminator
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(Original post by DoMakeSayThink)
Not quite sure what you mean by rule out the previous values, but because we know k > 3 > 1, we can just drop a 1 in, in place of the k, and it will strengthen the inequality. The way I originally phrased it might be a little more formal, but as long as you explain what you're doing, you're fine.
Okay, i think I understand. Its this strengthening the inequality bit which I wasn't understanding. But I see what is happening. Thanks a lot!
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TheRandomer
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Oh wow, I actually got something right in the maths forum
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The Bachelor
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(Original post by The Sherminator)
Its says: prove that  2^n > 2n  for    n >= 3

I said for n = k  2^k > 2k
n = k + 1  2^{k+1}  > 2k * 2
 2^{k+1} > 2k + 2k
The last statement you know for a fact. What about this claim:

 2^{k+1} > 2k + 2k > 2k + 2

Spoiler:
Show
And the rightmost side is, very conveniently, equal to 2(k+1) Which basically satisfies the inductive principle, don't you say?
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The Sherminator
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Yep, Runes, thanks.
It all makes sense now

TSR :love:
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silent ninja
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We need an FP3 thread. I remember that question, what year was it?

Some of the papers are tricky, others straightforward. June 02, I cant believe they give away 16 marks for two standard induction questions; 9 of which were for proving  2^{3n+2}+5^{n+1} is divisible by 3, whereas another paper I did had had a tougher induction and was worth almost half that. They must allocate marks randomly

There's a lot to remember in this unit, I have just realised! On another note, I've just figured out how to do matrices on my calculator. It's so nice being able to check those fiddly answers
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The Sherminator
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(Original post by silent ninja)
We need an FP3 thread. I remember that question, what year was it?

Some of the papers are tricky, others straightforward. June 02, I cant believe they give away 16 marks for two standard induction questions; 9 of which were for proving  2^{3n+2}+5^{n+1} is divisible by 3, whereas another paper I did had had a tougher induction and was worth almost half that. They must allocate marks randomly

There's a lot to remember in this unit, I have just realised! On another note, I've just figured out how to do matrices on my calculator. It's so nice being able to check those fiddly answers
Its a Solomon paper, not an Edexcel. I hate proof I just don't seem to get it, hopefully an easy one in the paper!
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silent ninja
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(Original post by The Sherminator)
Its a Solomon paper, not an Edexcel. I hate proof I just don't seem to get it, hopefully an easy one in the paper!
Yes I didnt quite understand it, until a thread by Glutamic turned up a few weeks back. The questions in the book are pretty good though-- only bad thing is that the method is crucial, and the answers dont give you this.
I felt the same for complex transformations until a thread a couple of weeks ago (that chapter also has few answers). Actually, I learn most of the useful maths stuff on here!

3 days to go. Crack on with them and i'm sure they'll make more sense. Post up if you're stuck and someone will be able to help. I think posting full solutions for these once someone has attempted them is the only way to see what's going on.
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The Sherminator
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Yep, good idea SN.
I got FP2 tomorrow, and then have to work on FP3 for a couple of days. Gah, so many maths exams and really close to each other. I wish the FP3 was on like Monday, I would have cracked it by then :teeth:
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