# Closed form solution equations involving finite sums

I am a beginner in Mathematica, so take this into account. I could not find anything similar in the internet so far.

I am trying to find a closed form solution for A in the equation below as a function of n, where n is countable and grows large. Potentially is infinite, but it would suffice to see a result for this when n is only large enough. I am using Reduce in the following way:

Reduce[
Sum[Subscript[λ, t]*((A + B)/2 + Subscript[b, t] - D - E)^2, {t, 1, n}] ==
Sum[Subscript[λ, t]*((B + C)/2 + Subscript[b, t] - D - E)^2, {t, 1, n}] &&
Sum[Subscript[λ, t], {t, 1, n}] == 1,
A]


Apparently Reduce is incapable of solving it. Perhaps I need another function.

When I use this specification above and assign numbers for n, for instance 3, the result is perfect and delivers what I need. Here is the example:

Reduce[Sum[
Subscript[λ, t]*((A + B)/2 + Subscript[b, t] - D - E)^2, {t, 1, 3}] ==
Sum[Subscript[λ, t]*((B + C)/2 + Subscript[b, t] - D - E)^2, {t, 1, 3}] &&
Sum[Subscript[λ, t], {t, 1, 3}] == 1,
A]


Here is the output -- it is in the closed form solution that I am looking for:

A ==
-2 B - C + 4 D + 4 E - 4 Subscript[b, 1] +
4 Subscript[b, 1] Subscript[λ, 2] -
4 Subscript[b, 2] Subscript[λ, 2] +
4 Subscript[b, 1] Subscript[λ, 3] -
4 Subscript[b, 3] Subscript[λ, 3]


Could you try to help me out with this?

-

$\sum _{t=1}^n l(t) \left(\frac{\text{A}+\text{B}}{2}+b(t)-\text{D}-\text{E}\right)^2-\sum _{t=1}^n l(t) \left(b(t)+\frac{\text{B}+\text{C}}{2}-\text{D}-\text{E}\right)^2=0$

$\sum _{t=1}^n (\text{A}-\text{C}) l(t) (\text{A}+4 b(t)+2 \text{B}+\text{C}-4 \text{D}-4 \text{E})=0$

Then

$\text{A}=\text{C}$

Or since $\sum _{t=1}^n l(t)=1$

$\text{A}=-4 b.l-2 \text{B}-\text{C}+4 \text{D}+4 \text{E}$

-
Thank you very much! It is exactly that form. – econ_sb 2 hours ago

I have rewritten your equations as

Sum[λ[t]*((a + b)/2 + β[t] - d - e)^2, {t, 1, n}] ==
Sum[λ[t]*((b + c)/2 + β[t] - d - e)^2, {t, 1, n}] &&
Sum[λ[t], {t, 1, n}] == 1


because identifiers C, D, and E are reserved for system use in Mathematica and because subscripts are a pain in computation.

I can't find a closed form solution for a, but by doing a little math I could derive a recursive definition.

a[1] = -2 b - c + 4 d + 4 e - 4 β[1];
a[n_Integer /; n > 1] := (a[n] = a[n - 1] + 4 β[1] λ[n] - 4 β[n] λ[n])


This, for example, gives

a[7]

-2 b - c + 4 d + 4 e - 4 β[1] +
4 β[1] λ[2] - 4 β[2] λ[2] + 4 β[1] λ[3] - 4 β[3] λ[3] +
4 β[1] λ[4] - 4 β[4] λ[4] + 4 β[1] λ[5] - 4 β[5] λ[5] +
4 β[1] λ[6] - 4 β[6] λ[6] + 4 β[1] λ[7] - 4 β[7] λ[7]


A little further analysis, using the relation Sum[λ[t], {t, 1, n}] == 1, gives

a[n_Integer?Positive] := 1 - 2 b - c + 4(d + e - Sum[β[t] λ[t], {t, 1, n}])


That is a far as I can carry the analysis, not knowing any details of β[t] and λ[t].

-
Thank you very much. Impressively helpful. – econ_sb 2 hours ago