Arithmetic Progressions | 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 $x$ .

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

Solution

Understand the Question

The house numbers are numbered consecutively from $1$ to $49$ , forming an Arithmetic Progression (AP) with first term $a = 1$ , common difference $d = 1$ , and total terms $n = 49$ .
Houses preceding house $x$ are numbered from $1$ to $(x - 1)$ , and their sum is given by $S_{x-1}$ .
Houses following house $x$ are numbered from $(x + 1)$ to $49$ , and their sum is obtained by subtracting the sum of the first $x$ houses from the total sum: $S_{49} - S_x$ .
We equate the two sums and solve the resulting quadratic equation for positive integer $x$ .

Step 1 · Set up the Equation

The house numbers form an AP: $1, 2, 3, \dots, 49$ , where $a = 1$ and $d = 1$ .The sum of first $n$ terms of an AP is $S_n = \dfrac{n}{2}[2a + (n-1)d]$

Given that the sum of houses preceding house $x$ equals the sum of houses following it $S_{x-1} = S_{49} - S_x$

Step 2 · Substitute and Solve for $x$

Substitute the formula for $S_n$ with $a = 1$ and $d = 1$ $\dfrac{x-1}{2} \left[2(1) + (x-1-1)(1)\right] = \dfrac{49}{2} \left[2(1) + (49-1)(1)\right] - \dfrac{x}{2} \left[2(1) + (x-1)(1)\right]$

Simplify within the brackets $\dfrac{x-1}{2}[x] = \dfrac{49}{2}[50] - \dfrac{x}{2}[x+1]$

Multiply by $2$ across the equation $(x-1)x = 49(50) - x(x+1)$

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

$x^2 - x = 2450 - x^2 - x$

$2x^2 - x = 2450 - x$

$2x^2 = 2450$

x^2 = \dfrac{2450}{2} = 1225

x = \sqrt{1225} = 35

Since house numbers are positive integers, $x = 35$ .

Answer

$x = 35$

Common Mistakes

Subtracting $S_{x-1}$ instead of $S_x$ : The houses following $x$ start from $x+1$ , so their sum is $S_{49} - S_x$ , not $S_{49} - S_{x-1}$ .
Negative Root: $\sqrt{1225} = \pm 35$ , but since $x$ represents a positive house number, discard $x = -35$ .

Question 4

More questions in Exercise 5.4