Puzzle – Where She Lives

Question:

It turned into a cocktail celebration in New York where I met Stephanie. We exchanged our telecell smartphone numbers and determined to fulfil every different soon. When she rang up and invited me to her residence that is how she gave the variety of her residence – ‘I stay on a protracted street. Numbered at the aspect of my residence are homes one, two, three, and so on. All the numbers on one aspect of my residence upload up similar to all of the numbers on the opposite aspect of my residence. I understand there are extra than fifty homes on that aspect of the street, however now no longer such a lot as 5 hundred’ Can you find Stephanie’s house number?

Solution:

Perfect Meridians are the numbers 1, 6, 35, 204, 1189, … because the sums of the remaining elements on each side (in their respective sequences) are equal. This means that the numbers of the houses on each side add up alike if the number of the house be 1, and there are no other houses, and if the number would be 6 with 8 houses in all, if 35 with 49 houses, if 204 with 288 houses if 1189 with 1681 houses and so on. 

Let’s see confirm it using the Number Theory.
We know that the summation formula for the first n Natural numbers is Sum = n(1 + n)/2
As the sequence of the first n Natural numbers contains a Perfect Meridian (M), then,
M = sqrt[n(n + 1)/2]
When n = 8: M = sqrt[8(8 + 1)/2] = 6
When n = 49: M = sqrt[49(49 + 1)/2] = 35
When n = 288: M = sqrt[288(288 + 1)/2] = 204
But, as per the question, there are more than 50 and less than 500 houses, so we are limited to a single case. The number of Stephanie’s house must be 204.