We have to prove that 1−r2n+11−r>(2n+1)r2
L.H.S. = Sum of a G.P. of 2n+1 terms whose first terms is 1, and common ratio r.
L.H.S. =1+r+r2+...+r2n
=(1+r2n)+(r+r2n−1)+(r2+r2n−2)+...n pairs +rn
Apply A.M.>G.M. on each pair
∴L.H.S.>(2rn+2rn+...+2rnn terms)+rn
=2n.rn+rn=(2n+1)rn