Number Systems

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 ______.

Let the function be defined by:

Define Let denote the number of solutions of the equation in the interval and Then the value of is .........