Question

Hølã Mátë§ !!

#No Spamming#

The houses in a row are numbered consecutively from 1 to 49. Show that there exist a value of X such that the sum of numbers of houses proceeding the house numbered X is equal to sum of the numbers of the houses following X.

CLASS - X
CH - 5
ARITHMETIC PROGRESSION ​

Answer 1

A/Q

AP:- (1 ,2 , 3.....x-1) ,x, (x+1 ,x+2......49)

Sn = n/2 (a+an)

$\frac{(x - 1)}{2} (1 + x - 1) = \frac{49 - x}{2} ( x + 1+ 49)$


$\frac{(x - 1)}{2} (x) = \frac{49 - x}{2} (x + 50)$


$\frac{ {x}^{2} - x}{2} = \frac{49x + 2450 - {x}^{2} - 50x }{2}$


$\frac{ {x}^{2} -x }{2 }= {\frac{ - {x}^{2} - x + 2450 }{2} }$


${x}^{2} - x = - {x}^{2} - x + 2450$


${x}^{2} + {x}^{2} -x + x = 2450$


$2 {x}^{2} = 2450$


Dividing by 2


${x}^{2} = 1225$

x=√35×35



$\implies \: \boxed{ \bf{x = 35}}$



${x}^{2} + x - 1225 = 0$


Solving by middle term splitting