Sorry in advance for posting a picture instead of the text, I'm not familiar with Latex yet...
My question is...
Why do we get: 4n^2 + 12n + 7 = 4((n − 1) + 1)^2 + 12((n − 1) + 1) + 7
And not this: 4n^2 + 12n + 7 = 4(n+1)^2 + 12(n+1) + 7
Asked
Active
Viewed 36 times
0
-
The first equation is trivially true since $(n-1)+1=n$. The second expression isn't set equal to anything, so I'm not sure what you are asking. – lulu Jan 07 '19 at 12:15
-
Sorry I just corrected it - It should just be n instead of k. I originally assumed a fresh witness n = k, but I left it out here to make it more simple. – Norton Jan 07 '19 at 12:16
-
I thought that when you add n+1 to 4n^2 it would be 4(n+1)^2, but that is not true I guess? – Norton Jan 07 '19 at 12:18
-
The second equation is obviously false. Just take $n=0$ for example. – lulu Jan 07 '19 at 12:20
-
Because $n=(n-1)+1$ and $n\neq n+1$. – drhab Jan 07 '19 at 12:21
-
Looking at the attached picture, the writer there is a bit careless. The argument shows that $P(n-1)\implies P(n)$ and NOT $P(n)\implies P(n+1)$ as stated. Of course, logically it is the same thing...but it is expressed poorly there. – lulu Jan 07 '19 at 12:34
-
For the LaTeX: literally wrap what you've written so far in dollar signs, and you're done. – user3482749 Jan 07 '19 at 12:39
1 Answers
0
The induction step should be "Assume that $P(n-1)$ holds, and then we show that $P(n)$ holds". This is an inaccuracy in the proof. Also, you should never say "Suppose we know that $P(n)$ is true", just say "Suppose that $P(n)$ is true". (That is the induction hypothesis, not the fact that we know it...)
A. Pongrácz
- 7,418
-
I thought the inductive step in a simple induction generally was to prove k+1 holds when having assumed a fresh witness of n = k. While the inductive step in strong induction often was to prove that p(n-1) holds for p(i) where i < n.
Is this a misunderstanding?
– Norton Jan 07 '19 at 12:27 -
There are different forms of induction. Also, it doesn't matter if you prove $P(n-1)\Rightarrow P(n)$ or $P(n)\Rightarrow P(n+1)$, just be consistent with your choice. – A. Pongrácz Jan 07 '19 at 12:29
-
The part where this assignment doesn't make sense for me, is that they want to show that P(n) implies P(n+1) and then write "4((n-1+1)^2" instead of "4(n+1)^2". – Norton Jan 07 '19 at 12:31
-
That is exactly what I put in my answer, please read it again. It is indeed an inaccuracy. – A. Pongrácz Jan 07 '19 at 12:34
-
Just to clarify that I understand it correctly. We CAN choose to say that P(n) implies P(n+1) is true instead of P(n-1), but then we would have "4(n+1)^2" instead of "4((n-1)+1)^2)"? – Norton Jan 07 '19 at 12:39
-