the proof is also simple IMHO.

You can define an indicator variable X that is 0 if heads and 1 if tails. The payoff is 10+ 10X or for N throws Sum(10+10X).The expected value is the E(Sum(10+10X)). which is 10N+10E(Sum(X)) which is 10N+10Sum(E(X)) =10N+10N/2=15N.