For each the following recursive sequences,(adsbygoogle = window.adsbygoogle || []).push({});

find a number K>0 such that [tex]{S_n}^2[/tex] + [tex]\bold{K}[/tex] is always composite.

1) [tex]S_0 = 10[/tex], [tex]S_1 = 11[/tex], [tex]S_n = 6S_{\left(n-1\right)}[/tex] - [tex]S_{\left(n-2\right)}[/tex]

2) [tex]S_0 = 14[/tex], [tex]S_1 = 17[/tex], [tex]S_n = 6S_{\left(n-1\right)}[/tex] - [tex]S_{\left(n-2\right)}[/tex]

Hint. Each problem has a separate K. Considering one of the series only, if you can find a second order equation in S_j and S_(j+1) that gives a constant for all j, and this equation also gives a second constant for the other series too, I would say that you are very near to solving these problems.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Prime-free sequence problems

**Physics Forums | Science Articles, Homework Help, Discussion**