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I'm given a growing list of a's and b's(size of a and b at the bottom) and had to form a recurrence equation for $a_{n}$ & $b_n$ (number of a's and b's respectively) and I was able to do it, these are what I got:

$a_n$=2$a_{n-1}$+$a_{n-2}$+$a_{n-3}$ and $b_n$=$b_{n-2}$+2$b_{n-3}$+3$b_{n-4}$

After doing this I needed to prove that $S_n$ = 2$S_{n-1}$+$S_{n-2}$, where $S_n$=$a_n$+$b_n$

However, when I tried solving for $S_n$, I get this:

$S_n$ = $S_{n-1}$+2$S_{n-2}$+2$S_{n-3}$+2$S_{n-4}$

When I try applying both of these $S_n$ they both get the correct answer, but idk how to simplify mine to be like the answer wanted

here are the numbers for as,bs etc

$a_0$=0, $a_1$=1, $a_2$=2, $a_3$=5 , $a_4$=13

and $b_0$=1, $b_1$=0, $b_2$=1, $b_3$=2, $b_4$=4

and $S_n$ being the sum of $a_n$ and $b_n$

Galbotrix
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  • This is hard to follow. It looks like you are using the same notation for different things, but it's all so vague that it is hard to tell. Are you trying to solve the recursion $S_n=2S_{n-1}+S_{n-2}$? But that's just a standard linear recursion. Are you trying to solve something else? – lulu Aug 02 '21 at 15:01
  • @lulu The $S_n$=2$S_{n-1}$+$S_{n-2}$ is what I need to "prove", by using the formulas that I obtained for $a_n$ and $b_n$ from the list of numbers at the end. I didn't know how to form the equation using $a_n$ and $b_n$ so I just tried solving it by using the list of S numbers (where $S_0$=$a_0$+$b_0$, ie 1=0+1, etc). Thats how I got $S_n$ = $S_{n-1}$+2$S_{n-2}$+2$S_{n-3}$+2$S_{n-4}$, but that's not the way it's asked for – Galbotrix Aug 02 '21 at 15:17
  • And we were meant to guess that? Please edit your post to ask a clear, coherent question. What are you given? What would you like to prove about it? – lulu Aug 02 '21 at 15:18
  • @lulu Sorry, I forgot to label $S_n$ properly first time. All I was given was the numbers for $a_{0,1,2,3,4}$ and $b_{0,1,2,3,4}$ and had to then proceed with the question. Let me know if that doesn't clear things up enough – Galbotrix Aug 02 '21 at 15:25
  • Is this even true? I get $a_5=2a_4+a_3+a_2=26+5+2=33$ and $b_5=b_3+2b_2+3b_1=2+0=4$ so $S_5=37$, right? Does that match the recursion? $S_4=17, S_3=7$ so $2S_4+S_3=41$. – lulu Aug 02 '21 at 15:28
  • @lulu You're right, I just double checked the given numbers and it only works for numbers 1,2,3,4 not 0. What method should I use to form $a_n$ and $b_n$? I was using equations where, for example when working on a would be: 5=2$D_1$+$D_2$ and repeat for the different values of a, plugging in known $a_{0,1,2,3,4}$ values in the $a_{n-i}$ slots and solving for the constant the multiplies by it – Galbotrix Aug 02 '21 at 15:35

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