can some1 chekck my working (1 Viewer)

[bos1234]

is this working out right? and what do i put for the last statment?

 

[PC]

Looks OK in a round about way.

I'd do B before A. It's just nicer that way.

When you do B, you've assumed n=k and proved true for n=k+1.
Then you do A. It works for n=1, so it must work for n=2.
It works for n=2, so it must work for n=3, and so on ... like dominoes.

 

[ssglain]

what do i put for the last statment?

As shown, the given result is true for n=k+1 if it is assumed true for n=k. Since it is proven true for n=1, it must be true for n=1+1=2. Therefore by mathematical induction, the result is true for all positive intergers n.

Induction is the absolute critical word in the final statement and it MUST be included. Marks will be deducted in the HSC if you don't 'by induction' in the final statement.

I personally prefer PC's order of proof. It's a much more logical approach. I don't quite understand why it is insisted that we must stick to Step 1: Prove true for n=1, Step 2: Assume true for n=k, Step 3: Prove true for n=k+1, Step 4: Conclusion. Do whatever your teacher says would be the best way as far as the HSC is concerned.

 

[buchanan]

what do i put for the last statment?

The statement is true for all positive integers n by the principle of mathematical induction.

The mantra "It works for n=1, so it must work for n=2.
It works for n=2, so it must work for n=3, and so on ..." is incorrect and is not to be used in HSC examinations according to the exam committee as discussed on page 9 of the following document

http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2005exams/pdf_doc/maths_ext_2_er_05.pdf

Thus the official BOS position on the issue is as follows:

The setting out of the induction warrants comment. A very large number of candidates who successfully completed the question (and many who attempted it) ended the induction proof with some version of the following:

"The statement is true for ​

n = 0 and hence is true for n = 1. The statement is true for n = 1and hence is true for n = 2 . The statement is true for n = 2 and hence is true for n = 3 and so on. Hence the statement is true for all integers n ≥ 0 (by induction)."

In a large number of cases the words .by induction. were omitted. Much time is wasted writing such a lengthy final statement and it would be better if candidates ended induction proofs with a simple statement like:

"Hence the statement is true for all n ≥ 0 by induction."​

This was discussed at length 2 years ago, but the issue arose again briefly at this year's examiners day because the 2006 MANSW HSC solutions contain the mantra in the Ext. 1 solutions, but not in the Ext. 2 solutions.

 

[bos1234]

The statement is true for all positive integers n by the principle of mathematical induction.

The mantra "It works for n=1, so it must work for n=2.
It works for n=2, so it must work for n=3, and so on ..." is incorrect and is not to be used in HSC examinations according to the exam committee as discussed on page 9 of the following document

http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2005exams/pdf_doc/maths_ext_2_er_05.pdf

Thus the official BOS position on the issue is as follows:

The setting out of the induction warrants comment. A very large number of candidates who successfully completed the question (and many who attempted it) ended the induction proof with some version of the following:

"The statement is true for
​

n = 0 and hence is true for n = 1. The statement is true for n = 1and hence is true for n = 2 . The statement is true for n = 2 and hence is true for n = 3 and so on. Hence the statement is true for all integers n ≥ 0 (by induction)."

In a large number of cases the words .by induction. were omitted. Much time is wasted writing such a lengthy final statement and it would be better if candidates ended induction proofs with a simple statement like:

"Hence the statement is true for all n ≥ 0 by induction."
​

This was discussed at length 2 years ago, but the issue arose again briefly at this year's examiners day because the 2006 MANSW HSC solutions contain the mantra in the Ext. 1 solutions, but not in the Ext. 2 solutions.

k thanks

i was going to put the statment above, "The statement is true for n = 2 and hence is true for n = 3 and so on. Hence the statement is true for all integers n ≥ 0 (by induction).""

but then i read the article..
The coroneos book concludes the proof like this.