For a natural number n , let α n = 19 n - 12 n . Then, the value of 31 α 9 - α 10 57 α 8 is ______

Question

For a natural number n , let &#945; n = 19 n - 12 n . Then, the value of 31 &#945; 9 - &#945; 10 57 &#945; 8 is ______

Aicharya · Accepted Answer

If $r_1$ and $r_2$ are roots of the quadratic $ax^2 + bx + c = 0$ then $r_1^n = 31r_1^{n-2} - 228r_1^{n-1}$ and $r_2^n = 31r_2^{n-2} - 228r_2^{n-1}$ So, $r_1^n - r_2^n = 31(r_1^{n-2} - r_2^{n-2}) - 228(r_1^{n-1} - r_2^{n-1})$ Or $a_n - 31a_{n-1} + 228a_{n-2} = 0$ where $a_n = r_1^n - r_2^n$ Since $19$ and $12$ are the roots of the quadratic $x^2 - 31x + 228 = 0$ we know, $a_n - 31a_{n-1} + 228a_{n-2} = 0$ Now, for $n = 10$, $\alpha_{10} - 31\alpha_{9} + 228\alpha_{8} = 0$ $\Rightarrow \frac{31\alpha_{9} - \alpha_{10}}{57\alpha_{8}} = 4$

For a natural number n , let α n = 19 n - 12 n . Then, the value of 31 α 9 - α 10 57 α 8…

Solution

For a natural number n , let &#945; n = 19 n - 12 n . Then, the value of 31 &#945; 9 - &#945; 10 57 &#945; 8…

Solution

For a natural number n , let α n = 19 n - 12 n . Then, the value of 31 α 9 - α 10 57 α 8…