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.
Top Questions on Number Systems
- If the HCF of two numbers is 15 and their product is 1800, find the LCM.
- If a man covers one-third distance at speeds 10 km/h, 20 km/h, and 60 km/h, find the average speed.
- Find the unit digit of \(1! + 2! + 3! + \cdots + 100!\).
- Find the unit digit of 1! + 2! + .... + 100!
- If the HCF of two numbers is 15 and the product is 1800. Find LCM.
Questions Asked in JEE Advanced exam
As shown in the figures, a uniform rod $ OO' $ of length $ l $ is hinged at the point $ O $ and held in place vertically between two walls using two massless springs of the same spring constant. The springs are connected at the midpoint and at the top-end $ (O') $ of the rod, as shown in Fig. 1, and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is $ f_1 $. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2, and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is $ f_2 $. Ignoring gravity and assuming motion only in the plane of the diagram, the value of $\frac{f_1}{f_2}$ is:
- JEE Advanced - 2025
- Waves and Oscillations
The reaction sequence given below is carried out with 16 moles of X. The yield of the major product in each step is given below the product in parentheses. The amount (in grams) of S produced is ____.
Use: Atomic mass (in amu): H = 1, C = 12, O = 16, Br = 80- JEE Advanced - 2025
- Organic Chemistry
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
- JEE Advanced - 2025
- binomial expansion formula
- The total number of real solutions of the equation $$ \theta = \tan^{-1}(2 \tan \theta) - \frac{1}{2} \sin^{-1} \left( \frac{6 \tan \theta}{9 + \tan^2 \theta} \right) $$ is
(Here, the inverse trigonometric functions $ \sin^{-1} x $ and $ \tan^{-1} x $ assume values in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and $(-\frac{\pi}{2}, \frac{\pi}{2})$, respectively.)- JEE Advanced - 2025
- Inverse Trigonometric Functions
Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is
- JEE Advanced - 2025
- Coordinate Geometry
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.