Pythagoras theorem states that in a right angled triangle the square of hypotenuse equals sum squares of other two sides.
A Pythagoras triple refers to a set of three positive integers a,b, and c that satisfy the condition:
a^2 + b^2 = c^2
To establish that a set of three integers forms a Pythagoras Triple we have to prove that the sum of squares of the two smaller integers is equal to the square of the square of the third number.
To find the value of n for which the set of numbers represented by (n, n+1, n+2) form a Pythagoras triple we form a equation based on on the condition of Pythagoras triple and then solve the equation for value of n.
n^ + (n+1)^2 = (n+2)^2
n^ + (n+1)^2 - (n+2)^2 = 0
n^ + n^2 + 2n + 1- n^2 - 4n - 4 = 0
n^ - 2n - 3 = 0
n^ - 3n + n - 3 = 0
n(n - 3) + 1(n - 3) = 0
(n + 1)(n - 3) = 0
Therefore n = 3 0r n = -1
Thus the condition of Pythagoras triple is satisfied for n = 3
To prove that the set of numbers represented by (n, n+1, n+3) cannot form a Pythagoras triple we form a equation based on on the condition of Pythagoras triple, solve the equation for value of n, and then show that these values of n are not integers
n^ + (n+1)^2 = (n+3)^2
n^ + (n+1)^2 - (n+3)^2 = 0
n^ + n^2 + 2n +1 - n^2 - 6n - 9 = 0
n^2 - 4n - 8 = 0
n^2 - 4n + 4 = 12
(n - 2)^2 = 12
n - 2 = 12^1/2 = 3.4641
n = 3.4641 + 2 = 5.4641
As only possible value of n is not an integer, the given set of number cannot form a Pythagoras triple.
Obviously the greatest side is the htpotenuse. So n+2 should be the sum of the squares on the other two sodes , n+1 and n forming the right angle. So,
(n+2)^2 = (n+1)^2+n^2. Or
(n^2+4n+4) = (n^2+2n+1)+n^2 . Or
0 = 2n2+2n+1 - (n^2+4n+4) . Or
0 = n^2-2n -3. Or
0 = (n+1)(n-3). Or
n-3 = 0 gives n =3. Therefore, 3 , 3+1= 4 and 3+2=5.
Let us assume that n^2+(n+1)^2 = (n+3)^2. Or
n^2 +n^2+2n+1= n^2+6n+9. Or
2n^2+2n- (n^2+6n ) = 9-1. Or
n^2-4n = 8 Or
n^2 -4n + 4 = 8+4 = 12 . Or
(n-2)^2 = sqrt12. Or
n = 2+2*3^(1/2).
Therefore , 2+2*3^(1/2), 3+2*3^(1/2) and 5+2*3(1/2) are the pytagorus triple, with the charecter, n, n+1 and n+3, but they are not the integers.