Let $$n \in N$$  and $$n < {\left( {\sqrt 2 + 1} \right)^6}.$$   Then the greatest value of $$n$$ is

Solution :
$$\eqalign{ & {\left( {\sqrt 2 + 1} \right)^6} = P + g = {\,^6}{C_0}{\left( {\sqrt 2 } \right)^6} + {\,^6}{C_1}{\left( {\sqrt 2 } \right)^5} + ..... + {\,^6}{C_6},\,{\text{where }}P = {\text{integer, }}0 < g < 1. \cr & {\left( {\sqrt 2 - 1} \right)^6} = f = {\,^6}{C_0}{\left( {\sqrt 2 } \right)^6} - {\,^6}{C_1}{\left( {\sqrt 2 } \right)^5} + .....,0 < f < 1 \cr & \therefore \,\,P + f + g = 2\left\{ {^6{C_0} \cdot {2^3} + {\,^6}{C_2} \cdot {2^2} + {\,^6}{C_4} \cdot 2 + {\,^6}{C_6}} \right\} \cr & P + f + g = 2\left( {8 + 15 \times 4 + 15 \times 2 + 1} \right) = 198 \cr & \therefore \,\,197 < P + g < 198 \cr & \therefore \,\,197 < {\left( {\sqrt 2 + 1} \right)^6} < 198 \cr & \therefore \,\,n < {\left( {\sqrt 2 + 1} \right)^6} \cr} $$
⇒ the greatest value of the natural number $$n = 197.$$