mathematical induction.

S(n) = a(1) + a(2) + ... + a(n - 1) + a(n);

Plato's conjecture: F(n) = [S(n)]^2;

basis: F(1) = [a(1)]^2 = [S(1)]^2 // yes, it holds.

induction: F(n) = F(n - 1) + S(n) * a(n) + S(n - 1) * a(n)

= [S(n - 1)]^2 + [S(n - 1) + a(n)] * a(n) + S(n - 1) * a(n)

= [S(n - 1) + a(n)]^2

= [S(n)]^2

Umm, our conjecture can be renamed as Plato's theorem now.

*Edited by author 18.04.2019 16:17*