Question

find three numbers in continued proportion such that their sum is 21 and product is 216​

Answer 1

Answer:

The three numbers in continued proportion are 3 , 6 , 12

Step-by-step explanation:

Given as :

The sum of three numbers in continued proportion = 21

The product of three numbers in continued proportion = 216

Let The three numbers in continued proportion = a , b , c

According to question

Continued proportion also know as mean proportion

And, For mean proportion ,

b² = a c             .........1

Since ,

sum of three numbers in continued proportion = 21

i.e  a + b + c = 21          ......2

product of three numbers in continued proportion = 216

i.e  a b c = 216            ......3

From 1 and 3

b² = $\dfrac{216}{b}$

i.e b³ = 216

∴   b = ∛216

or, b = 6

put the value of b into eq 3

So, ac = $\dfrac{216}{6}$

i.e ac= 36

Or, a = $\dfrac{36}{c}$

Put a and b value in eq 2

a + b + c = 21    

 $\dfrac{36}{c}$  + 6 + c = 21

or, $\dfrac{36}{c}$ + c = 21 - 6

Or. $\dfrac{36+c^{2} }{c}$ = 15

Or  c² - 15 c + 36 = 0

Or, c² - 12 c - 3  c + 36 = 0

or, c (c - 12) - 3 (c - 12) = 0

Or, (c - 12 ) ( c - 3) = 0

∴   c = 3 , c = 12

So, a = $\dfrac{36}{12}$ = 3

Hence, The three numbers in continued proportion are 3 , 6 , 12 Answer