SOLUTION: This one is kinda urgent..please i will appreciate ANY kind of help-
"Imagine that you are on a street
with houses marked 1 through n. There is a
house in between (x) such that
Question 694702: This one is kinda urgent..please i will appreciate ANY kind of help-
"Imagine that you are on a street
with houses marked 1 through n. There is a
house in between (x) such that the sum of
the house numbers to left of it equals the
sum of the house numbers to its right. If n is
between 50 and 500, what are n and x?" Answer by KMST(5328) (Show Source): You can put this solution on YOUR website! I do not see the whole elegant solution, but maybe this helps.
A sum of consecutive house numbers is the sum of an arithmetic sequence/progression, which can be calculated as the average of the first and last terms, times the number of terms.
The sum of the numbers of the houses to one side is
1+2+3+ ... +(x-2)+(x-1)=
The sum of the numbers of the houses to the other side is
(x+1)+(x+2)+ ... +(n-1)+n=
So --> --> --> -->
Since they are consecutive integers, and cannot have any common factors.
For their product to be twice a perfect square, we need two perfect squares with no common factors:
one perfect square will be a factor of ,
and another perfect square will be a factor of
Calling the larger one ,
either --> <--> ,
or --> ,
for some mutually prime positive integers and , with
whose squares must be between and
Since and , we should look at numbers from 5 to 22.
As the larger square is an odd number , is an odd number, and cannot be 22, so we should look at numbers from 5 to 22.
We are looking for a pair such that the square of the larger number is one more or one less than double the square of the smaller number.
We try numbers from 5 on as the smaller number , and see what works.
I tried them making a table, which is hard for me to render here, so I will just show the calculations. --> --> and =not a square --> --> =not a square and =not a square --> --> =not a square and =not a square --> --> =not a square and =not a square --> --> =not a square and =not a square --> --> =not a square and =not a square --> --> =not a square and =not a square --> --> =not a square and
The pair and with , works with ,
so --> --> -->
and --> --> --> -->
Since we expect only one solution, we could stop here.
Otherwise we would try further to see that nothing else works. --> --> =not a square and =not a square --> --> =not a square and =not a square --> --> =not a square and =not a square --> --> =not a number that could be or