If ${25}^{x-1}=5^{2x-1}-100$, find the value of $x$

(i) The distance between Sun and Saturn is $1433500000000 \mathrm{m}$.

(ii) The distance between Saturn and Uranus is $1439000000000 \mathrm{m}$.

(iii) The distance between Sun and Earth is $149600000000 \mathrm{m}$.

Represent the above distances in scientific notation and then find out which of these three distances is maximum?

Given that, 

$1^2+2^2=\frac{2\times 3\times 5}{6}, 1^2+2^2+3^2=\frac{3\times 4\times 7}{6}, 1^2+2^2+3^2+4^2=\frac{4\times 5\times 9}{6}$

Using the above pattern, determine ${\left(10\right)}^2+{\left(11\right)}^2+{\left(12\right)}^2+{\left(13\right)}^2+...+{\left(50\right)}^2$.

$1^2+2^2=\frac{2\times 3\times 5}{6}, 1^2+2^2+3^2=\frac{3\times 4\times 7}{6}, 1^2+2^2+3^2+4^2=\frac{4\times 5\times 9}{6}$etc.

Using the above pattern, determine ${\left(10\right)}^2+{\left(11\right)}^2+{\left(12\right)}^2+{\left(13\right)}^2+\dots +{\left(50\right)}^2$.