dan964
what
An essential skill. A shame none in 2014 hsc.
Methodology:
- Prove it is true for the FIRST CASE (which may not be n=1). If the FIRST CASE is trivial (example: 1+1 =2), then prove also for the
SECOND CASE.
- Assume the formula is true for n=k. Make your assumption then by letting n=k in the formula.
- Then for the next term usually n=k+1. You will need to 'rig' it to put your assumption back in, and prove it is of the same form as n=k.
Tips and Tricks:
- Don't do what some people do which is "Assume n=k". You are not proving that n=k. You are proving the formula is true.
- With inequalities, it is sometimes easier to see when rearranging your assumption by subtracting terms from both sides.
- Sometimes k>M, a certain value. If that is the case, it may come in handy.
- If the question involves a variable 'x' or a polynomial, then write A(x) or something not M.
Worked Examples. These are from my Year 11 book, as induction was the first topic we did in Year 11.
- Feel free to attempt them without looking at my proof.
For extension 2 students, if you are looking for more stuff; I would recommend proving something like Demoivre's Theorem.
The old 3U Fitzpatrick has some really good Mathematical Induction questions.
(The thing with Fitzpatrick is he puts all his hard questions at around Q2 sometimes)
View attachment 31612
Methodology:
- Prove it is true for the FIRST CASE (which may not be n=1). If the FIRST CASE is trivial (example: 1+1 =2), then prove also for the
SECOND CASE.
- Assume the formula is true for n=k. Make your assumption then by letting n=k in the formula.
- Then for the next term usually n=k+1. You will need to 'rig' it to put your assumption back in, and prove it is of the same form as n=k.
Tips and Tricks:
- Don't do what some people do which is "Assume n=k". You are not proving that n=k. You are proving the formula is true.
- With inequalities, it is sometimes easier to see when rearranging your assumption by subtracting terms from both sides.
- Sometimes k>M, a certain value. If that is the case, it may come in handy.
- If the question involves a variable 'x' or a polynomial, then write A(x) or something not M.
Worked Examples. These are from my Year 11 book, as induction was the first topic we did in Year 11.
- Feel free to attempt them without looking at my proof.
For extension 2 students, if you are looking for more stuff; I would recommend proving something like Demoivre's Theorem.
The old 3U Fitzpatrick has some really good Mathematical Induction questions.
(The thing with Fitzpatrick is he puts all his hard questions at around Q2 sometimes)
View attachment 31612
