Let the first term a and the common ratio r of a geometric...

Given that,

${a}^{2} + {\left(ar\right)}^{2} + {\left(a{r}^{2}\right)}^{2} = 33033, \left(a, r \in N\right)$

$⇒ {a}^{2}\left(1 + {r}^{2} + {r}^{4}\right) = {11}^{2} \times 273$

On comparing both sides of the equation we get,

$a = 11$ and $1 + {r}^{2} + {r}^{4} = 273$

$⇒ {r}^{2} + {r}^{4} = 272$

$⇒ {r}^{2}\left(1 + {r}^{2}\right) = 16 \times 17$

$⇒ r = 4$

Now $a + ar + a{r}^{2} = 11 + 11 \times 4 + 11 \times 16$

$= 11 + 44 + 176 = 231$.

Therefore, the required value is $231$