- Trending Categories
- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP
- Physics
- Chemistry
- Biology
- Mathematics
- English
- Economics
- Psychology
- Social Studies
- Fashion Studies

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

# The houses of a row are numbered consecutively from 1 to 49. Show that there is value of $x$ such that the sum of the numbers of the houses preceding the house numbered $x$ is equal to the sum of the numbers of the houses following it. Find this value of $x$.

**Given: **

The houses of a row are numbered consecutively from 1 to 49.

**To do:**

We have to show that there is value of $x$ such that the sum of the numbers of the houses preceding the house numbered $x$ is equal to the sum of the numbers of the houses following it and find this value of $x$.

**Solution:**

Let there be a value of $x$ such that the sum of the numbers of the houses preceding the house numbered $x$ is equal to the sum of the numbers of the houses following it.

This implies,

$1\ +\ 2\ +\ 3\ +\ .\ .\ .\ .\ +\ ( x\ –\ 1)=( x+1)+( x+2)+\ .\ .\ .\ .\ .\ + 49$

$\therefore 1+2+3+\ .\ .\ .\ .\ +( x–1)=[1+2+\ ....+x+( x–1)+....+49] \ –\ ( 1+2+3+.\ .\ .\ .\ +x$

Sum of the $n$ terms of the A.P. $S_{n} =\frac{n}{2}( a+l)$ [a is the first term and l is the last term of the given A.P.]

$\frac{x-1}{2}( 1+x-1) =\frac{49}{2}( 1+49) -\frac{x}{2}( a+l)$

$\therefore x( x\ –\ 1) \ =\ 49\ \times \ 50\ –\ x( 1\ +\ x)$

$\therefore x( x\ –\ 1) \ +\ x( 1\ +\ x) \ =\ 49\ \times \ 50$

$\therefore x( x-1+1+x) =49\times 50$

$2x^{2} = 49\times50$

$\therefore x^{2} =49\times 25$

$\Rightarrow x=\sqrt{49\times 25}$

$\therefore x=7\times 5=35$

Since $x$ is not a fraction, The value of $x$ satisfying the given condition exists and is equal to 35.

- Related Questions & Answers
- Find maximum N such that the sum of square of first N natural numbers is not more than X in Python
- Find maximum N such that the sum of square of first N natural numbers is not more than X in C++
- Maximum length of subarray such that sum of the subarray is even in C++
- How can I plot a histogram such that the heights of the bars sum to 1 in matplotlib?
- Convert the array such that the GCD of the array becomes 1 in C++
- PHP program to find the sum of cubes of natural numbers that are odd
- Maximum sum of distinct numbers such that LCM of these numbers is N in C++
- Count Triplets such that one of the numbers can be written as sum of the other two in C++
- Find a point such that sum of the Manhattan distances is minimize in C++
- Maximize the value of x + y + z such that ax + by + cz = n in C++
- Are media houses working on the news only for TRPs?
- Show that the set of all languages that are not recursively enumerable on {a} is not countable?
- Program to split a list of numbers such that the absolute difference of median values are smallest in Python
- Find maximum value of x such that n! % (k^x) = 0 in C++
- Check if a number from every row can be selected such that xor of the numbers is greater than zero in C++
- Find a permutation of 2N numbers such that the result of given expression is exactly 2K in C++