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in Arithmetic Progression by (36.5k points)
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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. And find this value of x. [Sx-1 = S49 – Sx]

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Given: Houses with numbers from 1 to 49. x is a number x such that, 

Sx-1 = S49 – Sx

⇒ x(x – 1) + x(x + 1) = 2 × 1225 

⇒ x2 – x + x2 + x = 2450 

⇒ 2x2 = 2450 

⇒ x2 = 2450/2 = 1225 

⇒ x = √1225 = ±35 

∴ x = 35 [∵ x is a counting number]

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