Question

Let $nϵN$ and $n<(\sqrt{2}+1{)}^6$,Then the greatest value of $n$ is

• A. $199$
• B. $198$
• C. $197$
• D. $196$

Answer 1

The correct option is C $197$

$\left[\right(\sqrt{2}+1{)}^2{]}^3$
$=(3+2\sqrt{2}{)}^3$
$=27+16\sqrt{2}+18\sqrt{2}(3+2\sqrt{2})$
$=27+16\sqrt{2}+54\sqrt{2}+72$
$=99+70\sqrt{2}$
$=197.99$
Since n has to be an integer
The greatest value of n which satisfies $n<(\sqrt{2}+1{)}^6$ is 197.