Question

Let denote the (r + 2) digit number where the first and the last digits are 7 and the remaining r digits are 5. Consider the sum S = 77 + 757 + 7557 + …+ . If S = , where m and n are natural numbers less than 3000, then the value of m + n is ______.

Answer 1

$S =77+757+7557+⋯+$
=$7(10+10^2+…+10^{99})+50(1+11+…+$$+7×99$

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

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

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

=$\frac{7 \times 10^{100}}{9} - \frac{70}{9} + \frac{50}{9}$$+7×99$

So, m+n=1219