Induction ODD/EVEN (1 Viewer)

[scaryshark09]

if an induction question says for odd/even positive integers:

should you just state that n = k is an odd/even number, and then let n = k + 2
OR
should you say n = 2k - 1 for odd, and then let n = 2k + 1
and n = 2k for even, and then let n = 2k + 2

 

[synthesisFR]

Most of the time n= k then n= k + 2 is enough for even or odd
Then if u need to u can let k = 2m or 2m +1 if the question requires it e.g factorising

 

[Luukas.2]

if an induction question says for odd/even positive integers:

should you just state that n = k is an odd/even number, and then let n = k + 2
OR
should you say n = 2k - 1 for odd, and then let n = 2k + 1
and n = 2k for even, and then let n = 2k + 2

It depends on how you view rigour and technicalities in proof.

Either approach will be fine in the HSC, so long as you do specify that k is odd or even (as appropriate) in stating the induction hypothesis when taking the first approach.

The second approach is mathematically preferable because of the nature of induction, but for reasons that are rarely spoken of at school level (and I suspect most teachers wouldn't know why).