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CONTINUED PROPORTION

__CONTINUED PROPORTION -__

**Continued proportion is a concept in mathematics where three quantities are said to be in continued proportion if the ratio between the first and the second quantity is the same as the ratio between the second and the third quantity.**

__Definition:-__

**If a, b, and c are three quantities, they are in continued proportion if:**

** a b**

** ------- = -------- **

** b c**

**In other words, the middle term b is called the mean proportional between the first term aaa and the third term c.**

__Formula:-__

** b² = a × c**

**For quantities a, b, and c to be in continued proportion:**

**This equation shows that b is the geometric mean of a and c.**

__Example:-__

**Let's say we have three numbers 4, 8, and 16.**

**To check if these numbers are in continued proportion:-**

** 1. Calculate the ratio of the first and second numbers:-**

** 4 1**

** -------- = --------**

** 8 2**

** 2. Calculate the ratio of the second and third numbers:-**

** 8 1**

** -------- = --------**

** 16 2**

**Since both ratios are equal, 4, 8, and 16 are in continued proportion.**

__Finding the Mean Proportional:-__

**If you are given a and c and need to find b, the mean proportional:-**

** a b**

** -------- = --------**

** b c**

**Or, a x c = b²**

**Or, b = √(a x c)**

**Example) Find the mean proportional between 9 and 16.**

** Ans.) As per the given condition, a = 9, c = 16, need to find value of b**

** b = √(a x c)**

**Or, b = √(9 x 16) = √144 = √12² = 12**

**So, 9, 12, and 16 are in continued proportion. (Ans.)**

__Applications:-__

**Continued proportion is useful in various mathematical contexts, including:**

__Geometry:-__In similar triangles, the sides of the triangles can be in continued proportion.__Algebra:-__Helps in solving problems involving geometric mean and sequences.__Proportional Reasoning:-__Used to compare relationships between different quantities.

__Summary:-__

**Continued proportion is a way to express that three quantities have a consistent relationship where the ratio of the first to the second is the same as the ratio of the second to the third. It highlights the concept of geometric mean and plays a significant role in various mathematical problems.**