Let 7 5 … 5 ⏞ 7 r denote the r + 2 digit number where the first and the last digits are 7 and…

Given,

$S=77+757+7557+\dots +\stackrel{98}{7\overbrace{5\dots 5}7}$

$\Rightarrow S=7\times 10+7+7\times 100+5\times 10+7+7\times 1000+5\times 100+5\times 10+7\dots +\stackrel{98}{7\overbrace{5\dots 5}7}$

$\Rightarrow S=7\left(10+{10}^2+\dots +{10}^{99}\right)+50\left(1+11+\dots +\stackrel{98}{\overbrace{111\dots 1}}\right)+7\times 99$

$\Rightarrow S=70\left(\frac{{10}^{99}-1}{9}\right)+\frac{50}{9}\left[\left(10-1\right)+\left({10}^2-1\right)+\dots +\left({10}^{98}-1\right)\right]+7\times 99$

$\Rightarrow S=70\left(\frac{{10}^{99}-1}{9}\right)+\frac{50}{9}\left[10\left(\frac{{10}^{98}-1}{9}\right)-98\right]+7\times 99$

$\Rightarrow S=\frac{7\times {10}^{100}}{9}-\frac{70}{9}+\frac{50}{9}\left[\frac{{10}^{99}-1-9}{9}-98\right]+7\times 99$

$\Rightarrow S=\frac{7\times {10}^{100}}{9}-\frac{70}{9}+\frac{50}{9}\left[\stackrel{99}{\overbrace{111\dots 1}}-99\right]+7\times 99$

$\Rightarrow S=\frac{7\times {10}^{100}-70+{\displaystyle \stackrel{99}{\overbrace{555\dots 5}}}0}{9}-550+693$

$\Rightarrow S=\frac{7{\displaystyle \stackrel{99}{\overbrace{555\dots .5}}}-70+143\times 9}{9}$

$\Rightarrow S=\frac{7{\displaystyle \stackrel{99}{\overbrace{55\dots 5}}}7+1210}{9}$

So, on comparing we get, $m+n=1210+9=1219$