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I'm trying to have Mathematica simplify and calculate the value of the first term of a sequence, defined recursively from an initial arbitrary n.

$P(n)=\frac{1}{2} (-V(n-1)+V(n)+R)$, where $V(n)$ is another arbitrary function, and $n$ is a natural number. The rest is defined by $P(k)=\frac{1}{2} \left(-\sum _{i=k+1}^n P(i)-V(k-1)+V(n)+R\right)$. I realise this is arbitrary, but I'd like to see if it simplifies to anything nice, and I need to run it for several functions $V$ and, ideally, arbitrary $n$.

I tried

RecurrenceTable[{P[k] == (V[n] + R - V[k - 1] - Sum[P[i], {i, k + 1, n}])/2, 
P[n] == (V[n] + R - V[n - 1])/2}, P, {k, n, 1}]

But I'm getting errors about recursion depth. Is there any way to do what I'm trying?

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Clear[P, k, n];
P[k_Integer?Positive, n_Integer?Positive] := 
 Piecewise[{{(V[n] + R - V[n - 1])/2, 
    k == n}, {(V[n] + R - V[k - 1] - Sum[P[i], {i, k + 1, n}])/2, 0 < k < n}}]

The first term (k == 1) for an arbitrary positive integer n is (V[n] + R - V[0] - Sum[P[k], {k, 2, n}])/2

Verifying for n up to 50

And @@ Table[P[1, n] == (V[n] + R - V[0] - Sum[P[k], {k, 2, n}])/2, {n, 50}]

(* True *)

Without specifying V and/or P it cannot be simplified further.

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