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?

#Algebra #Polynomials #PatternRecognition #Doubt

Easy Math Editor

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Comments

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.

Calvin Lin Staff - 7 years, 5 months ago

Thanks Calvin!

Juan rodrígez - 7 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$