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So I have the the sequence $a_1=2, a_2=8, a_3=50, a_4=344, a_n=8a_{n-1}-7a_{n-2} \text{ for $n$} \geq 3$ and I have to guess the closed form for $a_n$ and prove it by induction. My guess for the closed form was $a_n=1+7^{n-1}$. For the base case I choose $n=1$ and $n=2$ since $a_n=8a_{n-1}-7a_{n-2} \text{ for $n$} \geq3.$ Now, for my assumption I said that my conjecture was true for $n=k-1 \text{ and } n=k$ where $n \geq 1$ however I was wondering whether it should be $n \geq 3$ since we already know it's true for $n=1 \text{ and } n=2$ or if it actually doesn't matter whether it is $n \geq 3$ or $n \geq 1$ since they both work.

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    $\begingroup$ It does not matter, since as you said, they both work. But are you able to work through the induction case? $\endgroup$ – астон вілла олоф мэллбэрг Oct 20 '17 at 23:13
  • $\begingroup$ @астонвіллаолофмэллбэрг Yes. The induction case was straightforward but I just wasn't completely sure of my assumption. Thanks for clearing things up. $\endgroup$ – Hai Oct 21 '17 at 9:42

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