Let r 7 ⏞ 5 5 7 denote the (r + 2) digit number where the first and...

Given,

$S = 77 + 757 + 7557 + \ldots + \overset{98}{7\overset{⏞}{5 \ldots 5}7}$

$⇒ S = 7 \times 10 + 7 + 7 \times 100 + 5 \times 10 + 7 + 7 \times 1000 + 5 \times 100 + 5 \times 10 + 7 \ldots + \overset{98}{7\overset{⏞}{5 \ldots 5}7}$

$⇒ S = 7\left(10 + {10}^{2} + \ldots + {10}^{99}\right) + 50\left(1 + 11 + \ldots + \overset{98}{\overset{⏞}{111 \ldots 1}}\right) + 7 \times 99$

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

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

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

$⇒ S = \dfrac{7 \times {10}^{100}}{9} - \dfrac{70}{9} + \dfrac{50}{9}\left[\overset{99}{\overset{⏞}{111 \ldots 1}} - 99\right] + 7 \times 99$

$⇒ S = \dfrac{7 \times {10}^{100} - 70 + \overset{99}{\overset{⏞}{555 \ldots 5}}0}{9} - 550 + 693$

$⇒ S = \dfrac{7\overset{99}{\overset{⏞}{555 \ldots . 5}} - 70 + 143 \times 9}{9}$

$⇒ S = \dfrac{7\overset{99}{\overset{⏞}{55 \ldots 5}}7 + 1210}{9}$

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