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.
Question 1
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.
Thus:
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
Question 2
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
Thus:
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.
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