Petek said:Do you know about quadratic residues? If so, that's a hint.
The formula for determining when 1+2+...+n is a perfect square is: n(n+1)/2 = m^2, where n is the number of terms and m is the perfect square.
To prove that 1+2+...+n is a perfect square, you can use mathematical induction. First, show that the formula n(n+1)/2 is true for n=1. Then, assume it is true for n=k and prove it is true for n=k+1. This will prove that the formula holds for all natural numbers and therefore, 1+2+...+n is a perfect square.
No, there are no values of n for which 1+2+...+n is not a perfect square. This is because the formula n(n+1)/2 will always result in a perfect square number for any value of n.
Yes, for n=4, 1+2+3+4 = 10 which is a perfect square (3^2). Similarly, for n=10, 1+2+...+10 = 55 which is a perfect square (10^2).
Yes, this concept is used in various mathematical and engineering problems. For example, it can be used to determine the number of squares that can be formed in a grid pattern or to find the sum of consecutive odd or even numbers.