Hey Brilliant!
I was doing some zapping through techniques when I found that in Advanced Pattern Recognition:
What comes next:\( 3,7,13,21,31,43...?\)
**Oficial solution of Brilliant**

Since the 2nd difference of terms is the sequence ,\(2,2,2,2\).. this tells us that the the sequence can be generated by a polynomial of degree 2. In fact, this sequence is given by \(f(n)=n^{2}+n+2\), so we see that \(f(7)= 57\) which is indeed our guess.

Can someone explain me, why if the diference increases by two, it can be generated by a Polynomial of degree two, and how is that polynomial \(n^{2}+n+2\)found?

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## Comments

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TopNewestYou can read up on the technique Method of differences to understand why when the difference table is eventually constant, we have a polynomial. If you work through it, you will understand how to form the polynomial, from the initial conditions.

Alternatively, you can use a variety of polynomial-interpolation methods. The easiest of which, would be the Lagrange Interpolation Formula.

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Thanks Calvin!

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