/sci/ - Science & Math

>>10683160
Theorem: 10 is the biggest number
Proof:
Base case: n = 1 works because n < 10.
Induction case: Assume it works all the way up to k. We know that 1 < 10 and k < 10. Therefore k+1 < 10.

Therefore 10 is the biggest number.

>>10683160
Show that if it's true for k then it's also true for k+1.
Show it's true for a specific value of k.
Therefore it's true for all k>=that value.

>>10683208
Yea... btw there is actually another induction method, Cauchy Induction...
1) proof it works for some specific value k
2) proof if it works for n, it works for 2n
3) proof if it works for n, it works for n-1
Done.
I cant come up with an example atm but there do are some

>>10683160
CS majors belong in the >>>/g/hetto

>>10683167

have a series with an infinite number of terms.

Prove that if case n is true, then case n+1 is true
This means that the next term in the series will always share the condition assuming the previous does.

So if we can prove the condition true for the first term, we can effectively prove it true for all terms. Like knocking over a line of dominos.

Another form: prove that case n+1 is true if all the previous cases are true. This is harder, but it also works. So instead of just the previous term, you need to assume your condition for all previous terms.

The reason this is powerful is because you can assume your condition.

>>10683718
>Assumes true planar form now that the knowledge of induction is the true workhorse of all production.

>>10683160
How do we climb infinite stairs?
We start at the base, then we prove that if we are at the base, then we can move one step up.
Question: Prove that we can get to step 500
Base case: From the base of the stairway, I can move to step one by stepping.
Induction: Assume that it is true that for all levels lower than 500, I can reach the level. If I can reach these levels, then I can reach 499. Then, I can reach 500 by stepping.
Now we have proven that if it were true that we could reach a level below 500, then we could reach 500 by stepping. We proved that we could reach level 1. Take the base case, take the induction proof, and now we proved that we can reach 500. The thing we assumed, that we can reach all levels below 500, is the induction hypothesis. The induction hypothesis doesn't HAVE to be true; It just needs to help with your base case.

>>10683167
Kill yourself

>>10685816
1) We assumed that we could get to step 499.
2) We proved that if we could get to 499, we could get to 500. Therefore, we proved that if we can get to a step, we can get to a step above it.
3) We proved we could get to step one.
3a) Use the thing we prooved in step 2. Now, step by step, we have proven that we can get to any step we want

>>10683160
In Soviet Russia, induction proves YOU.

You prove if something holds for some arbitrary number k, then it holds for k+1. Then you prove it holds for 1, so it must hold for 2, and for 3, and for 4, and 5, and 6, and 7, etc.

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>>10683167
False. You forgot about negative and imaginary numbers.

>>10683160
Show true for n=1
Assume true for n=k
Prove true for n=k+1
Conclude that formula holds for n≥1
If you need me to provide an example I can.

>>10683167
>k < 10 implies k + 1 < 10
Wrong.