Školjka

%V0 Neka je $$ c_1 = 1,\;\;\;c_2 = 1 + \frac{1}{3},\;\;\ldots $$ $$ c_n = 1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \ldots + \frac{1}{2n - 1}. $$ Dokažite da vrijedi $$ c_n^2 + 2(c_n - c_1)^2 + 2(c_n - c_2)^2 + \ldots + 2(c_n - c_{n-1})^2 = n. $$