1. **Stating the problem:** We want to understand the concept of continued proportion, which is a sequence of numbers where each term is in proportion to the next. 2. **Definition:** Three or more numbers $a, b, c, \ldots$ are in continued proportion if the ratio of the first to the second equals the ratio of the second to the third, and so on. Formally, for three numbers $a, b, c$, they are in continued proportion if: $$\frac{a}{b} = \frac{b}{c}$$ 3. **Formula and explanation:** From the above equality, we can cross-multiply to get: $$a \times c = b^2$$ This means the middle term $b$ is the geometric mean of $a$ and $c$. 4. **Extending to more terms:** For four numbers $a, b, c, d$ in continued proportion: $$\frac{a}{b} = \frac{b}{c} = \frac{c}{d}$$ This implies each term after the first is obtained by multiplying the previous term by a common ratio $r$: $$b = a r, \quad c = a r^2, \quad d = a r^3$$ 5. **Important rules:** - The terms form a geometric progression. - The ratio between consecutive terms is constant. - The middle terms are geometric means between their neighbors. 6. **Example:** Suppose $a = 2$ and $c = 8$, find $b$ such that $a, b, c$ are in continued proportion. Using the formula: $$b^2 = a \times c = 2 \times 8 = 16$$ So, $$b = \sqrt{16} = 4$$ Therefore, the sequence is $2, 4, 8$. 7. **Summary:** Continued proportion means numbers are in geometric progression, each term after the first is multiplied by the same ratio, and the middle terms are geometric means of their neighbors.