Let the first term a and the common ratio r of a geometric progression be positive integers. If the sum of…

Given that,

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

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

On comparing both sides of the equation we get,

$a=11$ and $1+r^2+r^4=273$

$\Rightarrow r^2+r^4=272$

$\Rightarrow r^2\left(1+r^2\right)=16\times 17$

$\Rightarrow r=4$

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

$=11+44+176=231$.

Therefore, the required value is $231$