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# Pattern recognision

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?

Note by Juan Rodrígez
3 years, 3 months ago

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You 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. Staff · 3 years, 3 months ago