Answer:If 2 = 6
Then 0 = 4
Implies 0 = 1
Multiply both sides by any number and you shall be able to conclude that all numbers are but zero, including 9. This reduces the world of mathematics to absurdity.
Also, consider this case:
2 = 6
Implies 3 = 9
But the statement says 3 = 12.
Therefore 9 = 12.
I am just exploiting the inappropriate notation. But assume that you mean functions. Then consider this function:
f(n)=((n−1)(n−2)(n−3)(n−4)(n−5)(n−6)6!)(c−(n)(n+1))+n(n+1)
Where c is any arbitrary number. For the first six numbers, the given pattern shall follow, but what about the next one? The next one will yield c. And c is any arbitrary number that you select. Therefore you can use this relationship to generate any number you want for the seventh term, or extending it, we get:
f(n)=((n−1)(n−2)(n−3)(n−4)(n−5)(n−6)(n−7)(n−8)8!)(c−(n)(n+1))+n(n+1)
Where c is again, any arbitrary constant. Now you can select c to be root 2, or e or 1000000 or -3.23232424 or any number that you will. Interesting, isn't it.
The point I wish to make is that a finite number of cases cannot help you predict of what shall happen with the next one. Another case could be:
f(n)=n(n+1)(n−9)(n−9)
In this case, the 9th term would be undefined, however the pattern (n)(n+1) shall work for all other cases.
But then perhaps this does not answer your question, so let me just tell you that the simplest pattern possible can be found out by the method of polynomial regression. Use polynomial regression, and you shall get f(n) = n^2 + n, which is essentially n(n+1).
But this regression method would work only in cases that show polynomial behavior. What about other cases where the pattern is, let us say, exponential, or logarithmic, or rational (of the form polynomial divided by polynomial). The simplest way out would be to draw a graph and extend it. The question is, in which direction should you extend, which brings us back to the fact that finite number of cases cannot help us predict what shall happen with the next one.
Unfortunately, there is no mathematical answer to this question. The only one possible is through logical pattern matching, and a lot of people have answered it already
