Summation over 3 lists

I have three lists. As an example,

a = {10, 5, 6, 8, 7};
b = {4, 8, 9, 7, 9};
c = {7, 5, 12, 4, 1};


I want to get:

$$x(i)=\sum _{i=0}^i\frac{1}{2}\left(a_ib_i+a_{i+1}b_{i+1}\right)\left(c_{i+1}-c_i\right)$$ $i$ goes from $1$ to $4$, i.e. I want to get: $x(1)$,$x(2)$,$x(3)$ and $x(4)$.

So, I used the following code:

x=Sum[1/2*(a[[i]]*b[[i]] + a[[i + 1]]*b[[i + 1]])*(c[[i + 1]] -
c[[i]]), {i, 1, 4}]
For[i = 0, i < 4, x, Print[x]]


But, I didn't get expected output. Could anyone please tell me how to do this?

-

Table[Sum[1/2*(a[[i]]*b[[i]]+a[[i+1]]b[[i+1]])*(c[[i+1]]-c[[i]]), {i,1,k}],{k, 1, 4}]

-
Thank you so much. That is what I want. – TMH Apr 11 '13 at 22:52
Since I want to find $$y (i) = (Subscript[a, i + 1] - Subscript[a, i])/( Subscript[b, i + 1] - Subscript[b, i])$$ I tried with this code: Table[(a[i + 1] - a[i])/(b[[i + 1]] - b[[i]]), {i, 1, k}], {k, 1, 4} But I didn't get an error. Could you please tell me how to do this? – TMH Apr 12 '13 at 16:50
@TMH you seem to be confusing a[i] and a[[i]] in that code. Try: Table[Sum[(a[[i + 1]] - a[[i]])/(b[[i + 1]] - b[[i]]), {i, 1, k}], {k, 1, 4}] ; incidentially this looks rather nice if you select the expression and convert to StandardForm. – Mr.Wizard Apr 12 '13 at 18:27
@Mr.Wizard Thank you. It works. but, I don't want a sum. So, I removed it. I actually don't know how to convert the expression into standard form. Really appreciate your help on that too.Thanks again. – TMH Apr 12 '13 at 20:57
@TMH I see. Does the rest of the code work for you? You can make the conversion from the menu Cell > Convert To or with the keyboard shortcuts listed therein. – Mr.Wizard Apr 12 '13 at 23:48
Sum[1/2*(a[[i]]*b[[i]] + a[[i + 1]]*b[[i + 1]])*(c[[i + 1]] - c[[i]]), {i, 1, 4}]
(*
-(739/2)
*)


But I think this is better:

Tr[Differences@c (Most[a b] + Rest[a b])]/2

-
MovingAverage[a b, 2].Differences[c] – J. M. Apr 11 '13 at 16:12
@J.M. and belisarius Thank you for the answers. But, I want to get x values for each i values. i.e. I want to get values for x(1),x(2),x(3) and x(4) Could you please tell me how to do that? – TMH Apr 11 '13 at 22:09
@TMH thats the easy part: x[i_]:=MovingAverage[Take[a,i+1] Take[b,i+1], 2].Differences[Take[c,i+1])] for example. – Spawn1701D Apr 12 '13 at 6:47

Based on J. M.'s comment:

MovingAverage[a b, 2] * Differences[c] // Accumulate

{-80, 249, -191, -(739/2)}

-
Thank you Mr.Wizard. – TMH Apr 12 '13 at 1:01

Just a variant:

Needs["Developer"]
Accumulate[
`