Roots by Recursion

Olivia SutherlandRoots by RecursionBrief HistoryLeonhard Euler found a way to extract roots without use of the Quadratic formula or Cardanos formula but with recursion equations. This method is not as effective as it only finds an approximation and only one root.

Quadratic Formula

Rewrite equation

Recursion EquationsLet A = 1 and B = 2 (They can be any numbers)

=4 =1

C = 4B +A = 4(2) +1 = 9D = 4C +B = 4(9) + 2 = 38E = 4D +C = 4(38) +9 = 161F = 4E +D = 4(161) +38 = 682G = 4F +E = 4(682) + 161 = 2889

And this continues but Im going to stop there.

Compare From the Quadratic formula:

These values are only off by the 5th decimal point.

Now take the next term H:

The value is now only off by the 7th decimal point.

GeneralizeCan break up fraction by partial fraction decomposition

Geometric SeriesEach fraction on the right hand side can be put in a geometric series:

Matching TermsNow matching constant, linear, and quadratic terms:

Now for more general and larger exponents:

The RootRemember that p is the largest term

The root is so just take the reciprocal and you now have an approximation for the root.

The EndThank you for listening!

This factors to and the roots being and suppose that p is largest and is the smallest root.