# The houses of a row are numbered consecutively from 1 to 49. Show that there is 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. Find this value of x. [Hint S_{x} - 1 = S_{49} - S_{x} ]

**Solution:**

An arithmetic progression is a sequence has a common difference between any two of its consecutive numbers.

The sum of the first n terms of an AP is given by S_{n} = n/2 [2a + (n - 1) d], where a is the first term, d is the common difference and n is the number of terms.

The number of houses are 1,2,3, ..., 49

By observation, the numbers of houses are in an A.P.

Hence

- First-term, a = 1
- Common difference, d = 1

Let us assume that the number of x^{th} house can be expressed as below:

We know that sum of n terms in an A.P. is given by the formula S_{n} = n/2 [2a + (n - 1) d]

Sum of numbers of houses preceding x^{th} house = S_{x - 1}

S_{x - 1 }= (x - 1) / 2 [2a + ((x - 1) - 1)d]

= (x - 1) / 2 [2 × 1+ ( x - 2) × 1]

= (x - 1) / 2 [2 + x - 2]

= [x (x - 1)] / 2

By the given information we know that, Sum of number of houses following x^{th} house = S_{49} - S_{x}

S_{49} - S_{x} = 49 / 2 [2 × 1 + (49 - 1) × 1] - x / 2 [2 × 1 + (x - 1) × 1]

= 49 / 2 [2 + 48] - x / 2 [2 + x - 1]

= 49 / 2 [2 + 48] - x / 2 [2 + x - 1]

= 49 / 2 × 50 - x / 2 [x + 1]

= 1225 - [x (x + 1)] / 2

It is given that these sums are equal.

x (x - 1) / 2 = 1225 - x (x + 1) / 2

x² / 2 - x / 2 = 1225 - x² / 2 - x / 2

x^{2} = 1225

x = ± 35

As the number of houses cannot be a negative number, we consider the number of houses as x = 35

Therefore, house number 35 is such that the sum of the numbers of houses preceding the house numbered 35 is equal to the sum of the numbers of the houses following it.

**Video Solution:**

## The houses of a row are numbered consecutively from 1 to 49. Show that there is 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. Find this value of x.

### Class 10 Maths NCERT Solutions - Chapter 5 Exercise 5.4 Question 4:

The houses of a row are numbered consecutively from 1 to 49. Show that there is 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. Find this value of x.

The houses of a row are number consecutively from 1 to 49. Show that there is a value of x such that the sum of numbers of the houses preceding the house numbered x is equal to the sum of the number of houses following it. Then the value of x is 35. Therefore, house number 35 is such that the sum of the numbers of houses preceding the house numbered 35 is equal to the sum of the numbers of the houses following it