Just show that
...
.
Is the following an acceptable proof (be harsh )?
I mean, is the induction also required? Even if not, I need help with it:
- Base case - : ; define .
- Induction - : suppose .
Then ... how do I use the supposition?
Done that, just didn't write it down. The Halmos sign (with a "?" mark) indicates whether this is enough ... or is induction also needed (even if it is not, still help me with the induction step ).
MOD request: Can you please restore my spam-hijacked question?
You have given a very nice proof...
Suppose you were unaware of the proof you gave in your first post and were feeling lazy...
Here is the "Proof By Induction" method
The triangular numbers are 1, 3, 6, 10, 15, 21, 28,....
1=1^2
1+3=4=2^2
3+6=9=3^2 etc
P(k)
P(k+1)
Try to show that P(k) being valid (even if we don't know whether it is or not)
will cause P(k+1) to be valid.
Hence write P(k+1) in terms of P(k).
Proof
if P(k) really is valid
Now the line of dominoes is in place, hence you need to check if the first one falls.
true
true, depending on where you want to start from.