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$A(n) = A(n-1) + B(n-1)$

$B(n) = A(n-1)$

$A(1) = 2\ ,\ B(1) = 1 $

Please help to find closed form of $C(n) = A(n) + B(n)?$

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up vote 4 down vote accepted

You can write this as $$ \begin{pmatrix}A(n)\\B(n)\end{pmatrix} = \begin{pmatrix}1&1\\1&0\end{pmatrix} \cdot \begin{pmatrix}A(n-1)\\B(n-1)\end{pmatrix} $$ Now diagonalize that matrix to find its powers.

Or alternativey you can substitute the second equation into the first to obtain $A(n) = A(n-1) + A(n-2)$; now look up Fibonacci numbers.

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