As ciao comments code facilitates answers rather than pictures. As the base list is small I post the following. There will almost certainly be better ways and I have not simplified Sqrt:
dat = {{a1, b1, c1}, {a2, b2, c2}, {a3, b3, c3}, {a4, b4, c4}, {a5,
b5, c5}}
func[u_] := FoldList[Append, {u[[1]]}, u[[2 ;; -2]]]
col1 = Accumulate[Most@dat[[All, 1]]];
col2 = Sqrt[#.#] & /@ func[dat[[All, 2]]];
col3 = MapThread[#1/#2 &, {Rest@dat[[All, 3]],
MapThread[#1.#2/Plus @@ #2 &,
{func[dat[[All, 3]]], func[dat[[All, 1]]]}]}];
Transpose[{col1, col2, col3}]
this yields:
(*{{a1, Sqrt[b1^2], c2/c1}, {a1 + a2, Sqrt[
b1^2 + b2^2], ((a1 + a2) c3)/(a1 c1 + a2 c2)}, {a1 + a2 + a3, Sqrt[
b1^2 + b2^2 + b3^2], ((a1 + a2 + a3) c4)/(
a1 c1 + a2 c2 + a3 c3)}, {a1 + a2 + a3 + a4, Sqrt[
b1^2 + b2^2 + b3^2 + b4^2], ((a1 + a2 + a3 + a4) c5)/(
a1 c1 + a2 c2 + a3 c3 + a4 c4)}}*)