1. We are given the equation $x + \frac{1}{x} = 3$ and need to find the value of $x^{15} + \frac{1}{x^{15}}$. 2. Let's denote $a_n = x^n + \frac{1}{x^n}$. We know $a_1 = 3$. 3. We use the recurrence relation for $a_n$: $$ a_n = (x + \frac{1}{x})a_{n-1} - a_{n-2} $$ with $a_0 = 2$ (since $x^0 + \frac{1}{x^0} = 1 + 1 = 2$). 4. Use the recurrence to compute subsequent terms: - $a_0 = 2$ - $a_1 = 3$ For $n=2$: $$a_2 = 3 \cdot a_1 - a_0 = 3 \cdot 3 - 2 = 7$$ For $n=3$: $$a_3 = 3 \cdot a_2 - a_1 = 3 \cdot 7 - 3 = 18$$ For $n=4$: $$a_4 = 3 \cdot a_3 - a_2 = 3 \cdot 18 - 7 = 47$$ For $n=5$: $$a_5 = 3 \cdot a_4 - a_3 = 3 \cdot 47 - 18 = 123$$ For $n=6$: $$a_6 = 3 \cdot a_5 - a_4 = 3 \cdot 123 - 47 = 322$$ For $n=7$: $$a_7 = 3 \cdot a_6 - a_5 = 3 \cdot 322 - 123 = 843$$ For $n=8$: $$a_8 = 3 \cdot a_7 - a_6 = 3 \cdot 843 - 322 = 2207$$ For $n=9$: $$a_9 = 3 \cdot a_8 - a_7 = 3 \cdot 2207 - 843 = 5778$$ For $n=10$: $$a_{10} = 3 \cdot a_9 - a_8 = 3 \cdot 5778 - 2207 = 15127$$ For $n=11$: $$a_{11} = 3 \cdot a_{10} - a_9 = 3 \cdot 15127 - 5778 = 39603$$ For $n=12$: $$a_{12} = 3 \cdot a_{11} - a_{10} = 3 \cdot 39603 -15127 = 103682$$ For $n=13$: $$a_{13} = 3 \cdot a_{12} - a_{11} = 3 \cdot 103682 - 39603 = 271443$$ For $n=14$: $$a_{14} = 3 \cdot a_{13} - a_{12} = 3 \cdot 271443 - 103682 = 710647$$ For $n=15$: $$a_{15} = 3 \cdot a_{14} - a_{13} = 3 \cdot 710647 - 271443 = 1860498$$ 5. Therefore, the value of $x^{15} + \frac{1}{x^{15}}$ is $1860498$.