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For each the following recursive sequences,

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.

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.

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