Question

Divide 24 in three parts such that they are in AP and their product is 440.

Answer 1

To find the numbers in Arithmetic progression we divide the three terms of $a-d,a,a+d$ and equate with the value of $24$ and then we find the value of $a$ and after finding the value of $a$, we then find the value of $d$ which is found by the product of the three numbers which is equivalent to $440$.

Complete step by step solution:
Let us take the three numbers that are given in the question as $a-d,a,a+d$ which is divided to form the total value of $24$. So let us find the equation by placing the values equivalent to $24$ and then find the value of $a$. Placing the equation as:
$\Rightarrow \left( a-d \right)+a+\left( a+d \right)=24$
Removing the brackets, we get the value of the equation as:

$\Rightarrow a-d+a+a+d=24$
$\Rightarrow a+a+a=24$
$\Rightarrow 3a=24$
$\Rightarrow a=8$
Now that after placing the numbers together we have gotten the value of $a$ after placing it with the value of $24$, we now move to find the value of $d$ by multiplying the three terms and equating it with $440$, we get:
$\Rightarrow a-d\times a\times a+d=440$
$\Rightarrow a\left( {{a}^{2}}-{{d}^{2}} \right)=440$
Now placing the value of $a=8$, we get the equation as:
$\Rightarrow 8\left( {{8}^{2}}-{{d}^{2}} \right)=440$
$\Rightarrow d=\sqrt{9}$
$\Rightarrow d=\pm 3$
Now that we know the value of $a$ and $d$, placing the values in the three terms as $a-d,a,a+d$, we get the value of the three terms as:
$\Rightarrow a-d,a,a+d$
We have taken the value of $d=3$ as either can be taken.
$\Rightarrow 8-3,8,8+3$
$\Rightarrow 5,8,11$
Therefore, the value of the three numbers are given as, $5,8,11$.

Note:
The terms in arithmetic progression is written as: $...(a-2),(a-1),a,(a+1),(a+2)...$ where as for geometric progression it is written as $ar,a{{r}^{2}},a{{r}^{3}},....$ as in geometric progression, the terms move exponentially.