This is not a $G.P.,$ however, we can relate it to a $G.P.$ by writing the terms as

${S_n} = 7 + 77 + 777 + 7777 +  \ldots {\rm{ }}$ to $ n $ terms 

$ = \frac{7}{9}[9 + 99 + 999 + 9999 +  \ldots $ to $ n $ term $]$

$ = \frac{7}{9}[(10 – 1) + \left( {{{10}^2} – 1} \right) + \left( {{{10}^3} – 1} \right) + \left( {{{10}^4} – 1} \right) +  \ldots n{\rm{ }}$ term $]$

$=\frac{7}{9}\left[\left(10+10^{2}+10^{3}+\ldots n \text { terms }\right)-(1+1+1+\ldots n \text { terms })\right]$

$=\frac{7}{9}\left[\frac{10\left(10^{n}-1\right)}{10-1}-n\right]=\frac{7}{9}\left[\frac{10\left(10^{n}-1\right)}{9}-n\right]$