Question

if (x+1/x)=2, prove that (x^2+1/x^2)=( x^3+1/x^3)=(x^4+1/x^4)

Answer 1

Actually I'm going to prove that,

$\Large x^n+\dfrac{1}{x^n}=2\ \ \ \iff\ \ \ x+\dfrac{1}{x}=2\ \ \ \ \ \forall n\in \mathbb{Z}$

So, consider  x + 1/x = 2.

    x + 1/x = 2

Multiply both sides by x, so that,

    x² + 1 = 2x

Now, subtract 2x from both sides.

    x² + 1 - 2x = 0

⇒  x² - 2x + 1 = 0

On factorising the LHS, what will we get?!

    (x - 1)² = 0

Now, take the square root of both sides.

    x - 1 = 0

And finally we get,

    x = 1.

So, can't we substitute 1 instead of x?!

Okay, we see,

→   x + 1/x = 1 + 1/1 = 1 + 1 = 2

→   x² + 1/x² = 1 + 1/1² = 1 + 1 = 2

→   x³ + 1/x³ = 1 + 1/1³ = 1 + 1 = 2

→   x⁴ + 1/x⁴ = 1 + 1/1⁴ = 1 + 1 = 2

→   x⁵ + 1/x⁵ = 1 + 1/1⁵ = 1 + 1 = 2

That's why,

→   xⁿ + 1/xⁿ = 1 + 1/1ⁿ = 1 + 1 = 2

Hence proved that,

$\largex+\dfrac{1}{x}=x^2+\dfrac{1}{x^2}=x^3+\dfrac{1}{x^3}=x^4+\dfrac{1}{x^4}=\dots\! =x^n+\dfrac{1}{x^n}=2$

for every integer 'n'.