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  1. Let $A_{1}= 5, A_{2} = 7,$ $A_{n+1} = A_{n} + 6A_{n-1}$.

Find an explicit formula for $A_n$ in terms of $n$.

  1. Suppose $p_{n}$ is a sequence defined by $p_{1} = 0, p_{2} = 1$ & $\forall n$, $p_{n+1} = 2p_{n} - 2p_{n-1}$.

Find a formula for $p_{n}$ in terms of $n$.

How would I begin these problems?

user123
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  • One standard way is to write down the characteristic equation, find the roots. For the first it is $x^2-x-6=0$. The roots are $3$ and $-2$, so the general solution is $A\cdot 3^n+B\cdot (-2)^n$. Choose $A$ and $B$ so that the initial conditions are satisfied. – André Nicolas Apr 14 '15 at 18:14
  • make the ansatz $a_n=q^{n}$ – Dr. Sonnhard Graubner Apr 14 '15 at 18:19

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