Induction question-divisibility (1 Viewer)

[Fiction]

Question: n(n+2) is divisible by 4 if n is any even no.
Ans (without the test n=2 etc/first step) http://imgur.com/ZJiR8ux
I'm not understanding the last 2 lines in the answers. WHy is it that 4(p + k +2) transitions to 4p?
THanks in advance C:

 

[InteGrand]

Question: n(n+2) is divisible by 4 if n is any even no.
Ans (without the test n=2 etc/first step) http://imgur.com/ZJiR8ux
I'm not understanding the last 2 lines in the answers. WHy is it that 4(p + k +2) transitions to 4p?
THanks in advance C:

The question's just asking to show that the LHS is a multiple of 4. Once the answer's shown that it's equal to 4(p+k+2), it means a multiple of 4, since p+k+2 is an integer. It just transitions to 4p because "p+k+2" is just like "p", since "p" was just chosen to mean "an integer" basically (which p+k+2 is).

 

[FrankXie]

Looking from the context, the 4p in the last line was a typo, should be 4q.

 

[Fiction]

The question's just asking to show that the LHS is a multiple of 4. Once the answer's shown that it's equal to 4(p+k+2), it means a multiple of 4, since p+k+2 is an integer. It just transitions to 4p because "p+k+2" is just like "p", since "p" was just chosen to mean "an integer" basically (which p+k+2 is).

Looking from the context, the 4p in the last line was a typo, should be 4q.

Thank you!

 

[braintic]

I've never understood why these polynomial divisibility questions get asked as induction questions. Induction is not needed.

Since n is even, let n=2p where p is an integer.

n(n+2) = 2p(2p+2) = 4p(p+2)

Done. No inductive reasoning needed.

 

[Trebla]

I've never understood why these polynomial divisibility questions get asked as induction questions. Induction is not needed.

Since n is even, let n=2p where p is an integer.

n(n+2) = 2p(2p+2) = 4p(p+2)

Done. No inductive reasoning needed.

A lot of questions can be done with non-inductive methods but get asked to be proven by induction just for the purpose of practicing/familiarising the structure of an induction proof.

 

[braintic]

A lot of questions can be done with non-inductive methods but get asked to be proven by induction just for the purpose of practicing/familiarising the structure of an induction proof.

Can you think of any other common types of questions that are actually easier to do without induction that with?

 

[Carrotsticks]

Can you think of any other common types of questions that are actually easier to do without induction that with?

Pretty much all divisibility problems are much more easily proven directly.

Some of the proofs for closed forms of various partial sums can be done in a few lines.

 

[InteGrand]

Can you think of any other common types of questions that are actually easier to do without induction that with?

Divisibility questions, which can often quickly be solved by using the factorisation of .