1. **State the problem:** Given $|a|=5$, $|b|=20$, and $|a-b|=15.5$, find $|a+b|$. 2. **Recall the formula:** For any vectors or numbers $a$ and $b$, the identity $$|a+b|^2 + |a-b|^2 = 2(|a|^2 + |b|^2)$$ holds. 3. **Apply the formula:** Substitute the known values: $$|a+b|^2 + (15.5)^2 = 2(5^2 + 20^2)$$ 4. **Calculate the right side:** $$2(25 + 400) = 2 \times 425 = 850$$ 5. **Calculate the left side:** $$|a+b|^2 + 240.25 = 850$$ 6. **Isolate $|a+b|^2$:** $$|a+b|^2 = 850 - 240.25 = 609.75$$ 7. **Find $|a+b|$ by taking the square root:** $$|a+b| = \sqrt{609.75} \approx 24.69$$ **Final answer:** $$|a+b| \approx 24.69$$