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 ______.
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 ______.
Correct Answer: 1219
Approach Solution - 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
Approach Solution -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.
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Concepts Used:
Natural Numbers
Natural numbers are the set of positive integers (whole numbers greater than zero) that are used for counting and ordering. The set of natural numbers is denoted by the symbol N, and it includes all positive integers from 1 to infinity.
For example, the first few natural numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and so on. Natural numbers are used to count objects, measure time, and represent quantities such as age, weight, and length.
Also Read: Natural Numbers and Whole Numbers
Natural numbers have several important properties, including being closed under addition, subtraction, and multiplication. This means that when two natural numbers are added, subtracted, or multiplied, the result is always a natural number. However, natural numbers are not closed under division, as the quotient of two natural numbers may be a fraction or a decimal.
Natural numbers are a fundamental concept in mathematics and are used as the basis for many other number systems, including integers, rational numbers, and real numbers. They are used in many different fields, including science, engineering, and economics. The study of natural numbers is an important part of number theory, which is a branch of mathematics that deals with the properties of numbers and their relationships.