Five numbers are in A.P. whose sum is 25 and product is 2520, if one of these five numbers is $-\frac{1}{2}$ , then the greatest number amongst them is

Let terms be    $a-2d,a-d,a,a+d,a+2d$ 

Sum = $25\Rightarrow 5a=25\Rightarrow a=5$

  Also product = 2520     $\Rightarrow \left(5-2d\right)\left(5-d\right)5\left(5+d\right)\left(5+2d\right)=2520$

$\Rightarrow \left(25-4d^2\right)\left(25-d^2\right)=504$

$\Rightarrow 625-100d^2-25d^2+4d^4=504$

$\Rightarrow 4d^4-125d^2+625-504=0$

$\Rightarrow 4d^4-125d^2+121=0$

$\Rightarrow \left(d^2-1\right)\left(4d^2-121\right)=0$

$\Rightarrow d=\pm 1,d=\pm \frac{11}{2}$

$d=\pm 1$ is not possible since one of the terms is $-\frac{1}{2}$

$\Rightarrow d=\frac{11}{2}$

 $\therefore$Largest term = 5+ 2d = 5 + 11 = 16