Proving stuff (1 Viewer)

[Wohzazz]

I was taught to always write proofs in full.
Like for induction, in step four, i write like

If S(K) is true, S(k+1) has been proven true. Since S(1) is true, S(1+1)=S(2) is also true. Since S(2) is true, S(2+1)=S(3) is also true. Since S(3+1)=S(4) is also true and so on for all n, n >= 1

Is this like right and it is what they expect or can I shorten it to a

Hence, by the process of mathematical induction, statement is true for n, n>=1.

Also, for the inductions where you have to assume S(n) is true, for all integers n<=k, what sort of step 4 conclusion am I suppose to write

Also, i need clarification whether I could write (alt. angles) instead of alternate angles, AB//BC for example.
And for similar triangles, do I need to always write in triangles blah blah and blah blah before going into the prove.
And circle geo reasoning, do i need to ways name the arcs and circles, like

angle ACB = angle ADB (angles subtended from arc AB to the circumference of circle ACDB point C and D are equal)
-this whole reasoning again if there is a similar reasoning later on

Guess what i'm asking is what shortcuts can you take. Thanks for the answers.
Cheers

 

[cj_bridle]

i was told exactly the same as you... the induction i was told to write a full forth step like you demonstrated otherwise you could lose an easy mark for a poorly worded statement.. the last step is what lets the marker know you understand the concept of what an induction is..

same for circle geo. i was told not to abreviate anything..

 

[+Po1ntDeXt3r+]

well thats debatable..
for the forth step i think ures is a lil long.. i normally go for the generalised form after S(1) n S(2)
"If there is a k>=1, for which P(k) is true, then for this same k, P(k+1) is true by the principle of math. induction"
from memory goto jeff geha's 50 tips for the exact wording.. i didnt bring my maths books down to SA..

 

[gordo]

Since it is true for n=k+1 if it is true for n=k, therefore since it is true for n=1,then it is true for n = 1,2,3,4...

 

[withoutaface]

:. S(k) true implies S(k+1) true
But S(1) is true
:. by induction true for integers n>=1

Essentially just a variation on what others have already said.

 

[Supra]

yah i use wot justin jus sed, howeva our teacher didnt actually tell us that particular statement i dont htink, his statements used to b slightly incorrect, a relative taught me induction neway

 

[grimreaper]

In recent years, there has been no mark in the marking criteria for the last bit of an induction proof

 

[BillyMak]

In recent years, there has been no mark in the marking criteria for the last bit of an induction proof

hehe... I think I know the reason for that as well. People that had no idea were probably going "Step 1:, let n = 1...." doing the working and getting a mark, then not having a clue about the rest, then writing a concluding statement and getting another mark

 

[CrashOveride]

Are you sure about that? My trial had a mark for it and a lot of other trials seem to have a mark at the conclusion as well

 

[mervvyn]

On the topic of proving stuff in geometry....

when we use the alternate segment theorem can we just say (alt seg) or do we need to go through the whole "angle between a chord and a tangent" job...

 

[CrashOveride]

Yeah i was wondering about that. It seems we need to write out everything in full? I've always just written "alt. seg. theorem" ^^

 

[withoutaface]

Terry Lee said you can just write "alternate segment theory" as a proof.

 

[Wohzazz]

:. S(k) true implies S(k+1) true
But S(1) is true
:. by induction true for integers n>=1

Essentially just a variation on what others have already said.

So I take it, that is an acceptable step 4 under most circumstances. Looking at the mark allocation, there was a 2 marks induction q in q7 on one of the papers. A 1 minute step 4 makes you really want to shorten it esp if whole 1 only worth 2 or 3 marks.

About the circle geo thingy, I can see the mixed responces. So i can used generalised proofs like alt. seg theorem, angle subtended same arc are equal.

Also, say you are using the angle sum of a triangle reasoning. Do you need to write

angle ABC +angle BCA + angle CAB=180 (angle sum of triangle ABC)
angle ABC=90
angle BCA=45
therefore, angle CAB=180-90-45=45
or can it be shortened. Thank you for the replies

 

[withoutaface]

So I take it, that is an acceptable step 4 under most circumstances. Looking at the mark allocation, there was a 2 marks induction q in q7 on one of the papers. A 1 minute step 4 makes you really want to shorten it esp if whole 1 only worth 2 or 3 marks.

About the circle geo thingy, I can see the mixed responces. So i can used generalised proofs like alt. seg theorem, angle subtended same arc are equal.

Also, say you are using the angle sum of a triangle reasoning. Do you need to write

angle ABC +angle BCA + angle CAB=180 (angle sum of triangle ABC)
angle ABC=90
angle BCA=45
therefore, angle CAB=180-90-45=45
or can it be shortened. Thank you for the replies

I think for that you can just do:
CAB=180-BCA-ABC(angle sum of tri)
=180-90-45
=45

 

[grimreaper]

Are you sure about that? My trial had a mark for it and a lot of other trials seem to have a mark at the conclusion as well

Yes a lot of trials allocate a mark for it - but they just havent recently in the hsc (I'm not saying they definately wont this year though)

 

[McLake]

As said above, step 4 of induction in the past has NOT recvied a mark in the HSC, but this may change.

Induction step 4 should be long worded.

Circle Geo can be abriviated (there is a list in the 3U Maths syallabus which defines acceptable abriviations/symbols).

 

[CrashOveride]

Yes a lot of trials allocate a mark for it - but they just havent recently in the hsc (I'm not saying they definately wont this year though)

How do people know what the HSC allocates to it? Where did you find the BOS marking scheme

 

[McLake]

HSC Marking Notes. With the exams on the past HSC pages ...

 

[pinksneakers]

Terry Lee said you can just write "alternate segment theory" as a proof.

no, he tells us to write out the full thing cos some schools like mine deduct amrks if you dont write it all. same with induction.

 

[turtle_2468]

Not allocating a mark doesn't mean that you don't get a mark off if you just miss it altogether...
eg simplify root(15)-5/root(15)+5
If you make a mathematical error in a line where a mark isn't awarded, and then somehow you get back on the right track, ppl still have to take a mark off
You get the point.. I think...

With regard writing proof of induction: I'd just write the following...
3) Therefore if (*) holds for n=k, it holds for n=k+1
4) therefore by PMI (*) holds for all n.

With regard writing proofs in geometry:
It's always hard to judge how much to write. Even harder because many of the proofs are boring and long.. but anyway, I think you could probably get away with writing by alternate segment theorem if the tangent is in the preceding line. If you really can't be bothered writing something out again, label that line ... (1)
And then just write (1) as part of the reason for the theorem
eg
Therefore <ABC=<BDE (Alt. segt. Thm., from (1), (2))