From natural numbers 1 to 288, find the numbe | vedclass

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(C) Let the required number be $x.$ The sum of natural numbers smaller than $x$ is given by the sum of integers from $1$ to $x-1.$This sum is $S_1 = \

Vedclass · Accepted Answer

$204$

Vedclass · Answer

(C) Let the required number be $x.$ The sum of natural numbers smaller than $x$ is given by the sum of integers from $1$ to $x-1.$This sum is $S_1 = \frac{(x-1)x}{2}.$The sum of natural numbers greater than $x$ up to $288$ is the sum of integers from $x+1$ to $288.$This sum is $S_2 = \sum_{i=1}^{288} i - \sum_{i=1}^{x} i = \frac{288 	imes 289}{2} - \frac{x(x+1)}{2}.$According to the problem,$S_1 = S_2.$$\frac{x^2 - x}{2} = \frac{288 	imes 289}{2} - \frac{x^2 + x}{2}.$Multiplying by $2,$ we get $x^2 - x = 288 	imes 289 - x^2 - x.$$2x^2 = 288 	imes 289.$$x^2 = 144 	imes 289.$$x = \sqrt{144 	imes 289} = 12 	imes 17 = 204.$Thus,the required number is $204.$

From natural numbers $1$ to $288,$ find the number $x$ such that the sum of all natural numbers smaller than $x$ is equal to the sum of all natural numbers greater than $x$ but less than or equal to $288.$

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