Let α and β are roots $$x2$$−$$7.32x$$−$$3=0$$ and let $$An=$$α$$n+$$βn, then $$A37$$−$$3A35A36$$ is equal to

$$An+1=($$α$$+$$β$$)$$α$$n+$$βn−αβαn−$$1+$$βn−$$1$$ $$=7.32An+3An$$−$$1$$ ⇒ $$A37$$−$$3A35A36=7.32$$

Theory of expressions

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Question

Let $α$ and $β$ are roots $x^{2} - 7 .32 x - 3 = 0$ and let $A_{n} = α_{n} + β_{n}, then \frac{A_{37} - 3 A_{35}}{A_{36}}$ is equal to

Moderate

Solution

$\begin{matrix} A_{n + 1} = (α + β) (α^{n} + β^{n}) - αβ (α^{n - 1} + β^{n - 1}) \\ = 7 .32 A_{n} + 3 A_{n - 1} \end{matrix}$
$\Rightarrow \frac{A_{37} - 3 A_{35}}{A_{36}} = 7 .32$

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