Why dont we square on both sides or in other words use (k_n + k_n-1 + k_1 +k_2)^2 = 2*((k_n)^2 + (k_n-1)^2 + (k_1)^2 + (k_2)^2)...and then solve recursively.
By the way this question was like the "Mother Of Simplification". I couldnt believe my own eyes when it boiled down to such a small formula which is kn=k4 + (n-4)*((k4-k3) + (n-3)*(k1+k2)) where n>4.