If x a - y b -1 and x 2 a 2 - y 2 b 2 - ab a-b then prove that x n-1 a - y n-1 b - ab a-b n -

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Exponential Representation for Irrational Numbers

If x a + y ...

Question

If $\frac{x}{a} + \frac{y}{b} = 1$ and $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = \frac{a b}{a + b}$ then prove that $\frac{x^{n + 1}}{a} + \frac{y^{n + 1}}{b} = {(\frac{a b}{a + b})}^{n} .$

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x + y = a + b, x^{2} + y^{2} = a^{2} + b^{2}

x^{n} + y^{n} = a^{n} + b^{n}

x = \frac{a}{a + b}

y = \frac{b}{a - b}

\frac{1}{x} + \frac{1}{y} = \frac{a^{2} + b^{2}}{a b}

x = 1 + a + a^{2} + . . . . .

y = 1 + b + b^{2} . . . .

| a | < 1

| b | < 1

1 + a b + a^{2} b^{2} + . . . . = \frac{x y}{x + y - 1}

s i n^{- 1} \frac{2 a}{1 + a^{2}} - c o s^{- 1} \frac{1 - b^{2}}{1 + b^{2}} = {t a n}^{- 1} \frac{2 x}{1 - x^{2}}

x = \frac{a - b}{1 + a b}

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\frac{a^{2}}{x^{2}} - \frac{b^{2}}{y^{2}} = 1

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