1. **Problem:** Find the sum of the first 4 terms $S_4$ of a geometric progression where the first term $b_1=4$ and the common ratio $q=\frac{1}{2}$. 2. **Formula:** The sum of the first $n$ terms of a geometric progression is given by $$S_n = b_1 \frac{1-q^n}{1-q}$$ This formula applies when $q \neq 1$. 3. **Substitute values:** Here, $n=4$, $b_1=4$, and $q=\frac{1}{2}$. $$S_4 = 4 \times \frac{1-(\frac{1}{2})^4}{1-\frac{1}{2}}$$ 4. **Calculate powers and simplify:** $$(\frac{1}{2})^4 = \frac{1}{16}$$ So, $$S_4 = 4 \times \frac{1-\frac{1}{16}}{1-\frac{1}{2}} = 4 \times \frac{\frac{15}{16}}{\frac{1}{2}}$$ 5. **Divide fractions:** $$\frac{\frac{15}{16}}{\frac{1}{2}} = \frac{15}{16} \times 2 = \frac{30}{16} = \frac{15}{8}$$ 6. **Multiply by $b_1$:** $$S_4 = 4 \times \frac{15}{8} = \frac{60}{8} = 7.5$$ **Final answer:** $$\boxed{7.5}$$