Word Count : 476

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हमारे पास प्रारंभिक शर्तें हैं: [ a_0 = 2, \quad a_1 = 5 ] और पुनरावृत्ति संबंध: [ a_n = 6 a_{n-1} - 5 a_{n-2} + 1 \quad \text{के लिए } n \ge 2. ] सबसे पहले, हम संबंधित समानांगीय समीकरण पर ध्यान देते हैं (असमानांगीय भाग को छोड़कर): [ a_n - 6 a_{n-1} + 5 a_{n-2} = 0 ] इसका चरित्र समीकरण है: [ r^2 - 6r + 5 = 0 ] इसे हल करते हैं: [ r^2 - 6r + 5 = 0 \implies r = \frac{6 \pm \sqrt{36 - 20}}{2} = \frac{6 \pm \sqrt{16}}{2} = \frac{6 \pm 4}{2} ] तो हमें दो मूल मिलते हैं: [ r_1 = \frac{6 + 4}{2} = 5, \quad r_2 = ________ ___ ______ ______ ______ _____.
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