Question

three numbers are in continued proportion and the middle number is 24.if the sum of the first and the third is 52 find the number​

Answer 1

Answer:

Three terms are x, y, z

Therefore,

x +z = 52

y = 24

Here, x-z = 24

Therefore,

x= 38

y= 24

z= 14

Here, y^2 = xz

Answer 2

We need to recall the following definition for the proportion.

If $a, b, c$ are in continued proportion, then $ac=b^{2}$

This problem is about the proportion and quadratic equation.

Given:

Sum first and the third term in continued proportion is $52$.

Middle term is $24$

Let the three numbers be $p,q,r$

As they are in continued proportion, we get

$pr=q^2$

$pr=(24)^2$

$pr=576$

⇒  $r=\frac{576}{p}$           ........$(1)$          

Also, it is given that

$p+r=52$           ........$(2)$

Substitute equation $(1)$ in $(2)$

$p+\frac{576}{p} =52$

⇒ $p^{2}+576=52p$

⇒ $p^{2}-52p+576=0$

⇒ $(p-36)(p-16)=0$

⇒  $p=36$  or  $p=16$

∴ $r=16$  or  $r=36$

Hence, the numbers are $16,24,36$  or $36,24,16$