For a natural number n, let an=19n-12n. Then, the value of 31α9-α1057α8 is

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For a natural number n, let $a_{n} = 19^{n} - 12^{n}$ . Then, the value of $\frac{31 α_{9} - α_{10}}{57 α_{8}}$ is

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answer is 4.

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Detailed Solution

$a_{n} = 19^{n} - 12^{n} \Rightarrow \frac{31 α_{9} - α_{10}}{57 α_{8}} = \frac{31 (19^{9} - 12^{9}) - (19^{10} - 12^{10})}{57 α_{8}} \Rightarrow \frac{31 α_{9} - α_{10}}{57 α_{8}} = \frac{19^{9} (31 - 19) - 12^{9} (31 - 12)}{57 α_{8}} \Rightarrow \frac{31 α_{9} - α_{10}}{57 α_{8}} = \frac{19^{9} \cdot 12 - 12^{19} \cdot 19}{57 α_{8}} \Rightarrow \frac{31 α_{9} - α_{10}}{57 α_{8}} = \frac{12 \cdot 19 (19^{8} - 12^{8})}{57 α_{8}} = 4$