I tried this last night and i'm looking at it this mornng. I'm still stuck on a bit of it:
My method is like this:Observe that:
1=1
1-4=-(1+2)
1-4+9=1+2+3
1-4+9-16=-(1+2+3+4)
Guess the general law and prove it by induction.
For n=1:
Hence true for n=1.
Assume true for n=k:
For n=k+1:
At this point it's clear that the LHS does not equal the right RHS. This implies that my initial expression for the sum is incorrect. However, it works for n=1 and n=2 so I think it's right.
Can anyone see what i'm doing wrong?
Hello, Showcase_22!
We have: .
Verify . . . True
Assume
Add to both sides.
. .
We must show that the right side is the right side of
. . which looks like this: .
The right side is: .
Factor: .
Factor: .
. . And we have: .
The inductive proof is complete.
I would like to give a different way for the proof.
It is not by induction, its by direct computation.
This may be useful for other problems.
We want to prove that
for any .
I will only consider the case where is odd because the case where is even is very similar.
Since is odd, we may find such that holds.
....................
....................
....................
....................
....................
....................
....................
Hence the proof for this case is completed.
Let be even and pick to satisfy .
Then, we have
.
The rest is similar.
Oh WOW!
I wasn't expecting quite so much help. I was trying it myself and I was still getting a bit stuck on it (my typo when I wrote was really throwing me).
Anyway, thanks a bundle. I'm going to attempt this question and solve it like there's no tomorrow!!!
After trying and SUCESSFULLY solved the problem: I used a method similar to Soroban's except I did it a little differently:
After a lot of working you end up with the required result (edit: I just noticed this is exactly what Soroban did. Rather than add it to both sides I added it to the same side).
I like your method bkarpuz. Direct computation is a method I haven't heard of before. It took me a few read-throughs but I finally got it. =D