gmatkkvinu wrote:1 and 8 are the first two natural number for which 1+2+3+.........+n is a perfect square.which number is the fourth such number?
I still maintain that this is out of scope for the GMAT, but here's my solution nevertheless. . .
We might start with this formula:
1+2+3...+n = (n)(n+1)/2
So, we're looking for values of n such that
(n)(n+1)/2 is a perfect square (i.e., the square of an integer).
As the question shows, when n =
1, we get
(n)(n+1)/2 = 1, which is a perfect square.
Similarly, when n =
8, we get
(n)(n+1)/2 = 36, which is a perfect square.
IMPORTANT: First notice that n and n+1 are consecutive integers. Also notice that something nice happens when n = 8. When we divide n (8) by 2, we get 4 (which is a perfect square) and n+1 (9) is already a perfect square.
So, at this point, we need only check consecutive integers where one of the integers is already a square and the other integer,
when divided by 2, becomes a square.
Check 15 and 16: 16 is a square, but 15/2 is not a square. Keep checking.
Check 16 and 17: 16 is a square, but 17/2 is not a square. Keep checking.
Check 24 and 25: 25 is a square, but 24/2 is not a square. Keep checking.
Check 25 and 26: 25 is a square, but 26/2 is not a square. Keep checking.
IMPORTANT: From now on, I won't check squares that are even, because the number that's 1 greater will be ODD, and odd/2 cannot be a perfect square. Likewise, the number that's 1 less will be ODD, and odd/2 cannot be a perfect square.
Check 48 and 49: 49 is a square, but 48/2 is not a square. Keep checking.
Check 49 and 50: 49 is a square, AND 50/2 is a square. So, 49 is the 3rd such number.
Check 80 and 81: 81 is a square, but 80/2 is not a square. Keep checking.
Check 81 and 82: 81 is a square, but 82/2 is not a square. Keep checking.
Check 120 and 121: 121 is a square, but 120/2 is not a square. Keep checking.
Check 121 and 122: 121 is a square, but 122/2 is not a square. Keep checking.
Check 168 and 169: 169 is a square, but 168/2 is not a square. Keep checking.
Check 169 and 170: 169 is a square, but 170/2 is not a square. Keep checking.
Check 224 and 225: 225 is a square, but 224/2 is not a square. Keep checking.
Check 225 and 226: 225 is a square, but 226/2 is not a square. Keep checking.
Check 288 and 289: 289 is a square, AND 288/2 is a square. So, n = 288 is the 4th such number.
There may be a faster (simpler) way, but that's my solution.
Cheers,
Brent
