Question

If a, b, c are in continued proportion, then which of the following to follow?
$
  {\text{A}}{\text{. }}{{\text{b}}^2}{\text{ = ac}} \\
  {\text{B}}{\text{. c is the third proportional}} \\
  {\text{C}}{\text{. }}\dfrac{{\text{a}}}{{\text{b}}} = \dfrac{{\text{c}}}{{\text{b}}} \\
  {\text{D}}{\text{. }}\dfrac{{\text{a}}}{{\text{b}}} = \dfrac{{\text{d}}}{{\text{c}}} \\
$

Answer 1

Hint: Three quantities are said to be in continued proportion; if the ratio between the first and the second is equal to the ratio between the second and the third.

Complete step-by-step answer:

Given Data,
a, b, c are in continued proportion i.e. a : b :: b : c.
⟹$\dfrac{{\text{a}}}{{\text{b}}} = \dfrac{{\text{b}}}{{\text{c}}}$
⟹${\text{ac = }}{{\text{b}}^2}$
⟹${{\text{b}}^2} = {\text{ac}}$
Here, b is called the mean proportional of a and c, as the square of the middle term is equal to the product of 1st term and 3rd term.
Also, if a : b :: b : c, then c is called the third proportional of a, b.
Hence, Options A and B are correct.

Note: In order to solve this type of question the key is to understand the concept of continued proportion. Then we apply the concept to all the given options to verify if the conditions hold true, then we determine the answer.