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# 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 Sₓ ₋ ₁ = S₄₉ - Sₓ ]

**Solution:**

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

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

The number of houses is 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 Sₙ = n/2 [2a + (n - 1) d]

Sum of number of houses preceding x^{th} house = Sₓ ₋ ₁

Sₓ ₋ 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 ---------- (1)

By the given information we know that, sum of number of houses following x^{th} house = S₄₉ - Sₓ

S₄₉ - Sₓ = 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) × 50 - (x / 2) (x + 1)

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

It is given that these sums are equal, that is equation (1) = equation(2)

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

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

On solving further we get,

x^{2} = 1225

x = ± 35

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

**☛ Check: **NCERT Solutions Class 10 Maths Chapter 5

**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

**Summary:**

The houses of a row are numbered consecutively from 1 to 49. We have shown 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. The value of x is 35.

**☛ Related Questions:**

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