Question

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-, =S49-S]​

Answer 1

$\large\underline{\sf{Solution-}}$

As it is given that the required house number is '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.

It implies,

$\rm :\longmapsto\:1 + 2 + 3 + - - + (x - 1) = (x + 1) + (x + 2) + - - + 49$

Now,

Consider,

$\rm :\longmapsto\: (x + 1) + (x + 2) + - - + 49$

can be rewritten as

$\rm \: = \:(1 + 2 + - - + 49) - (1 + 2 + 3 + - - + x)$

We know,

$\boxed{ \tt{ \: 1 + 2 + 3 + - - - + n = \frac{n(n + 1)}{2} \: }}$

So, using this, we get

$\rm \: = \:\dfrac{49(49 + 1)}{2} - \dfrac{x(x + 1)}{2}$

Now,

Consider,

$\rm :\longmapsto\:1 + 2 + 3 + - - - + (x - 1)$

$\rm \: = \:\dfrac{(x - 1)(x - 1 + 1)}{2}$

$\rm \: = \:\dfrac{(x - 1)(x - 1 + 1)}{2}$

$\rm \: = \:\dfrac{(x - 1)x}{2}$

So,

$\red{\rm :\longmapsto\:1 + 2 + 3 + - - + (x - 1) = (x + 1) + (x + 2) + - - + 49}$

$\rm :\longmapsto\:\dfrac{(x - 1)x}{2} = \dfrac{49 \times 50}{2} - \dfrac{x(x + 1)}{2}$

$\rm :\longmapsto\:x(x - 1) = 49 \times 50 - x(x + 1)$

$\rm :\longmapsto\: {x}^{2} - x = 49 \times 50 - {x}^{2} - x$

$\rm :\longmapsto\: 2{x}^{2} = 49 \times 50$

$\rm :\longmapsto\: {x}^{2} = 49 \times 25$

$\rm :\longmapsto\:x = \sqrt{49 \times 25}$

$\bf\implies \:x = 7 \times 5 = 35$

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

Additional Information :-

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

↝ Sum of n  terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• Sₙ is the sum of n terms of AP.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.