Divide 32 into 4 parts which are in AP such that the ratio of the product of the extremes to the product of the means is 7:5 .
If I'm interpreting your problem correctly, something is wrong in your problem.
Consider the sequence:
a-3d,a-d,a+d,a+3d
This is an arithmetic progression with constant difference 2d.
The product of the 2 outer terms is a^2 - 9d^2. The product of the 2 inner terms is a^2 - d^2.
Your requested ratio is:
$\displaystyle \frac{a^2-9d^2}{a^2-d^2} = \frac {7}{5}$
which leads to
$\displaystyle -38d^2 = 2a^2$
which is impossible.
Hello, prantik007!
Please check the problem for typos.
As stated, the situation is patently impossible.
Divide 32 into 4 parts which are in AP such that the ratio of the product
of the extremes to the product of the means is 7:5 .??
The four parts are: .$\displaystyle a,\;a+d,\;a+2d,\;a+3d$
The product of the extremes is: .$\displaystyle P_E \:=\:a(a+3d) \:=\:a^2 + 3ad$
The product of the means is: .$\displaystyle P_M \:=\:(a+d)(a+2d) \:=\:a^2 + 3ad + 2d^2$
. . It is obvious that: .$\displaystyle P_E \:<\:P_M$
So how can . $\displaystyle P_E:P_M \:=\:{\color{red}7:5}$ ?