$\text{Given ,}f(x+y)=f\left(x\right)+f\left(y\right)$

$f\left(1\right)=10$

$\therefore f\left(2\right)=f(1+1)=f\left(1\right)+f\left(1\right)=10+10=20$

$\text{Similarly }f\left(3\right)=30,f\left(4\right)=40, \text{and so on.}$

$\therefore f\left(n\right)=10n$

$\sum _1^n{\left(f\left(x\right)\right)}^2=\sum _1^n{\left(10n\right)}^2$

$\Rightarrow 100\times \frac{n\times (n+1)\times (2n+1)}{6} \because \sum _{}^{}{n}^2=\frac{n\times (n+1)\times (2n+1)}{6}$

$\Rightarrow \frac{50}{3}n(n+1)(2n+1)$