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

Correct Answer: 1219

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

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

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

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

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

So, m+n=1219
 

Answer 2

The sum consists of numbers of the form \( 7\ldots 7 \) followed by \( 5\ldots 5 \). The number of digits increases by one in each term.

First, we observe the pattern in the numbers. Each term can be written as a sum of geometric series. For example:

• 77 = 7 * 10 + 7
• 757 = 7 * 10^2 + 5 * 10 + 7
• 7557 = 7 * 10^3 + 5 * 10^2 + 5 * 10 + 7
• ... so on.

The general term for the \( k \)-th number in the sum is:

\(T_k = 7 \times 10^{k} + 5 \times (10^{k-1} + 10^{k-2} + \cdots + 10 + 1) + 7\)

This can be simplified and represented as a geometric series, which can be summed to find the value of \( S \).

The sum \( S \) then evaluates to:

\(S = m \times n\)

After solving this equation for \( m \) and \( n \), we find that:

The value of \( m + n \) is 1219.