1. **Stating the problem:** We have a sequence $\{a_n\}$ with the first term $a_1 = 2$ and the recurrence relation $$a_{n+1} = \frac{a_n}{2} + \frac{1}{2n - 1}.$$ We need to find $a_4$. 2. **Formula and approach:** The recurrence relation tells us how to get the next term from the current term. We will calculate $a_2$, $a_3$, and then $a_4$ step-by-step. 3. **Calculate $a_2$:** $$a_2 = \frac{a_1}{2} + \frac{1}{2(1) - 1} = \frac{2}{2} + \frac{1}{1} = 1 + 1 = 2.$$ 4. **Calculate $a_3$:** $$a_3 = \frac{a_2}{2} + \frac{1}{2(2) - 1} = \frac{2}{2} + \frac{1}{3} = 1 + \frac{1}{3} = \frac{4}{3}.$$ 5. **Calculate $a_4$:** $$a_4 = \frac{a_3}{2} + \frac{1}{2(3) - 1} = \frac{\frac{4}{3}}{2} + \frac{1}{5} = \frac{4}{3} \times \frac{1}{2} + \frac{1}{5} = \frac{2}{3} + \frac{1}{5}.$$ 6. **Simplify $a_4$:** Find common denominator 15: $$a_4 = \frac{2}{3} + \frac{1}{5} = \frac{2 \times 5}{15} + \frac{1 \times 3}{15} = \frac{10}{15} + \frac{3}{15} = \frac{13}{15}.$$ **Final answer:** $$a_4 = \frac{13}{15}.$$