How are formulas with inductive proofs discovered?

[Avichal]

There are certain formulas for which only inductive proofs are known. But since we need to know the formula first to prove it using mathematical induction, how do they get the formula in the first place?
Here is an example: - Euler's formula in graph theory states that v-e+f=2 for all planar graphs. I think only an inductive proof is currently known. So how did he come up with this formula without proving it first?...guessing?

 

[Ryuzaki]

Here is some interesting information (courtesy of micromass):-

www.homepages.math.uic.edu/~kauffman/DCalc.pdf

www.math.upenn.edu/~wilf/gfologyLinked2.pdf

I believe the topic of generatingfunctionology answers yours question, though I'm no expert.

 

[Vargo]

There are different ways.
1. Try to find a formula empirically. For example, if you want to know a formula for the nth triangular number 1+2+3+...+n, you could compute the first 20 and try to find a relationship with n by examining the numbers. You can notice a lot of patterns just be examining data closely.

2. Sometimes you can know a lot about a problem without completely understanding it. In those cases you might have reason to believe something but not be able to prove it. For example, you might suspect that ln(n) - 1 -1/2 -1/3 - ... - 1/n approaches a limit without being able to prove it. Basically, even if you don't understand something very well, you can try to apply induction. You can "get lucky" with induction by proving something that you don't understand that well. The drawback is that such proofs don't always add much to your understanding or suggest new avenues to pursue.

I suspect that empirical observation is the main way that people conjecture formulas before they understand them.

 

[chiro]

Hey Avichal.

One should note the important step in an inductive proof which is the delta between successive values of n.

It obviously depends on the nature of the constraint (summation, multiplication, inequality, etc) but looking at the delta between steps is a good way to proving something or at least getting an idea of whether it potentially could be true (in the context of statements in induction proofs).

 

[CellCoree]

There are several ways in which formulas with inductive proofs can be discovered. One way is through experimentation and observation. Mathematicians may notice a pattern or relationship between certain numbers or objects and then use mathematical induction to prove it for all cases. This is often seen in combinatorics and number theory.

Another way is through deductive reasoning. Mathematicians may start with a known formula or theorem and use deductive reasoning to derive a new formula, which can then be proven using mathematical induction. This is a common approach in algebra and geometry.

It is also possible that a formula is discovered through a combination of experimentation and deductive reasoning. For example, a mathematician may notice a pattern in a series of numbers and use deductive reasoning to come up with a general formula, which can then be proven using mathematical induction.

In the case of Euler's formula in graph theory, it is likely that Euler observed a pattern in different planar graphs and used deductive reasoning to come up with the formula v-e+f=2. This formula was then later proven using mathematical induction.

In short, formulas with inductive proofs can be discovered through a combination of experimentation, deductive reasoning, and observation. It is a process that requires creativity, intuition, and a deep understanding of mathematical concepts.