we want to get all possible values with $a_1$

Question

I would appreciate it if somebody could help me with the following problem:

Given $$a_{n+1}=\begin{cases} \dfrac{a_n}{2}& (a_n \: \text{even})\\ n{a_n}& (a_n \: \text{odd})\end{cases},\:a_8= 10$$ we want to get all possible values with $a_1$. How can I get it using Mathematica11? Please.

Answer 1

We could invert the functions and go "backwards". So, a[n-1] could be 2*a[n] if 2*a[n] is even (which it will always be, so we always have at least one possibility). But also, a[n-1] could be a[n]/(n-1) if that value is odd.

So, given a value for an index, we can compute possible previous values:

PreviousAs[{idx_Integer, value_Integer}] :=
  With[
    {evenCase = 2 value, oddCase = value/(idx - 1)},
    Thread[{idx - 1, {evenCase, If[OddQ[oddCase], oddCase, Nothing]}}]]

We're given a value for index 8, and we encode this as {8,10}. We want to go backwards 7 steps, so evaluate this to get the possible values for index 1:

Nest[Catenate@*Map[PreviousAs], {{8, 10}}, 7]

We can test that it works (although this test doesn't guarantee that we found them all--maybe my logic above was incomplete) by defining the "forward" function:

NextA[{idx_Integer, value_Integer?EvenQ}] := {idx + 1, value/2};
NextA[{idx_Integer, value_Integer?OddQ}] := {idx + 1, idx value}

and iterating on one of the candidates generated above:

Nest[NextA, <...copy/paste one of the candidates here...>, 7]

Hopefully we get {8,10} back.

Answer 2

We first write the recurrence equation like: a[n_]:= ... and set a[1]=a1 where a1 is not yet known:

Clear[a];
a[1] = a1;
a[n_] := a[n] = If[OddQ[n], a[n - 1]/1, a[n - 1] (n - 1)];

Now we can get a[8]:

a[8]
(* 105 a1 *)

From this we see that a1= 10/105:

a1=10/105;

To test:

a[8]
(* 10 *)

and the whole sequence up to 10:

Table[a[i],{i,10}]
(* {2/21, 2/21, 2/21, 2/7, 2/7, 10/7, 10/7, 10, 10, 90} *)