Q4:

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 : $S_{x-1} = S_{49} - S_x$]

Given: A row of houses numbered consecutively from $1$ to $49$. Let the house number in question be $x$.

To Find: The value of $x$ such that the sum of house numbers preceding $x$ is equal to the sum of house numbers following $x$.

Step 1: Defining the Sums

The house numbers form an Arithmetic Progression (AP) where the first term $a = 1$ and the common difference $d = 1$.

The sum of the first $n$ terms of an AP is given by the formula: $S_n = \frac{n}{2}(a + l)$, where $l$ is the last term.

The sum of all house numbers from $1$ to $49$ is:

$S_{49} = \frac{49}{2}(1 + 49) = \frac{49}{2}(50) = 49 \times 25 = 1225$

Step 2: Formulating the Equation

Let the sum of houses preceding $x$ be $S_{x-1}$.

$S_{x-1} = \frac{(x-1)}{2}(1 + (x-1)) = \frac{(x-1)x}{2}$

The sum of houses following $x$ is the total sum minus the sum up to $x$:

Sum following $x = S_{49} - S_x$

Where $S_x = \frac{x(x+1)}{2}$.

According to the problem, $S_{x-1} = S_{49} - S_x$.

Step 3: Solving for $x$

Substitute the expressions into the equation:

$\frac{x(x-1)}{2} = 1225 - \frac{x(x+1)}{2}$

Multiply the entire equation by $2$ to clear the denominators:

$x(x-1) = 2450 - x(x+1)$ $x^2 - x = 2450 - (x^2 + x)$ $x^2 - x = 2450 - x^2 - x$

Add $x$ to both sides:

$x^2 = 2450 - x^2$

Add $x^2$ to both sides:

$2x^2 = 2450$

Divide by $2$:

$x^2 = 1225$

Take the square root of both sides (since $x$ must be positive):

$x = \sqrt{1225}$ $x = 35$

Step 4: Verification

Sum of houses preceding $35$ ($1$ to $34$):

$S_{34} = \frac{34}{2}(1 + 34) = 17 \times 35 = 595$

Sum of houses following $35$ ($36$ to $49$):

$S_{49} - S_{35} = 1225 - \frac{35(36)}{2} = 1225 - (35 \times 18) = 1225 - 630 = 595$

Since $595 = 595$, the value is verified.

Final Answer: The value of $x$ is $35$.