Mathematics: Apythagorean triple is a triple of natural numbers satisfying the equation a^2+b^2+c^2. one way to produce a pythagorean triple is to add the reciprocals of any two consecutive even or odd numbers. for example, 1/5+1/7=12/35. now 12^2+35^2=1369. this is a pythagorean triple if 1369 is a perfect square, which it is since 1369=37^2. so 12, 35, 37 is a pythagorean triple. prove that this method always works.

Apythagorean triple is a triple of natural numbers

Ответ:

binodkharal2048

31.12.2021

X, x+2 - two consecutive odd or even numbers
Add the reciprocals of these numbers.
$\frac{1}{x}+\frac{1}{x+2}=\frac{x+2}{x(x+2)}+\frac{x}{x(x+2)}=\frac{x+2+x}{x^2+2x}=\frac{2x+2}{x^2+2x}
$

Now add the squares of the numerator and denominator, as in the example.
$(2x+2)^2+(x^2+2x)^2= \\ 4x^2+8x+4+x^4+4x^3+4x^2= \\ x^4+4x^3+8x^2+8x+4$

So this number has to be a perfect square.
$x^4+4x^3+8x^2+8x+4= \\
x^4+2x^3+2x^2+2x^3+4x^2+4x+2x^2+4x+4= \\
x^2(x^2+2x+2)+2x(x^2+2x+2)+2(x^2+2x+2)= \\
(x^2+2x+2)(x^2+2x+2)= \\
(x^2+2x+2)^2$
It is a perfect square, so this method always works.

The numbers $2x+2, \ x^2+2x, \ (x^2+2x+2)^2$ are a Pythagorean triple for any $x \in \mathbb{N^+}$ .

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Ответ:

lizzy2951

16.02.2022

Mathematics

C. (1, - 1)

Hope this will help

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