Find the sum

$\text {Let , } S_n=7+77+777+7777+....\text {(n terms)} \\ \implies S_n=7(1+11+111+1111+....\text {n terms}) \\ \implies S_n=\dfrac79(9+99+999+9999+....\text {n terms}) \\ \implies S_n=\dfrac79[(10-1)+(100-1)+(1000-1)+(10000-1)+....\text {n terms}] \\ \implies S_n=\dfrac79[(10+10^2+10^3+\cdots\text{n terms})-(1+1+1+\cdots\text{n terms})] \\ S_n=\dfrac79\left(\dfrac{10(10^n-1)}{10-1}-n\right)\\ S_n=\dfrac{7}{81}[10(10^n-1)-9n] \\ \text{Put n}=13 \\ S_{13}=\dfrac{7}{81}\left(10(10^{13}-1)-9\cdot13\right) = \boxed{8641975308631}$

$\text{: Generalized for a Digit (N) :} \\ \text {Let , } S_n=N+NN+NNN+NNNN+....\text {(n terms)} \\ \implies S_n=N(1+11+111+1111+....\text {n terms}) \\ \implies S_n=\dfrac{N}{9}(9+99+999+9999+....\text {n terms}) \\ \implies S_n=\dfrac{N}{9}[(10-1)+(100-1)+(1000-1)+(10000-1)+....\text {n terms}] \\ \implies S_n=\dfrac{N}{9}[(10+10^2+10^3+\cdots\text{n terms})-(1+1+1+\cdots\text{n terms})] \\ S_n=\dfrac{N}{9}\left(\dfrac{10(10^n-1)}{10-1}-n\right)\\ S_n=\dfrac{N}{81}[10(10^n-1)-9n] \\$