The sum and product of five consecutive terms of an A.P. are 25 and 2520 respectively. If one of the term is $\frac{1}{2}$, then the value of the greatest term is :

Let the terms of the A.P. be  $a-2d,a-d$,$a,a+d,a+2d$. Then,

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

Product of terms = 2520

$\begin{array}{l}\Rightarrow a\left(a^2-4d^2\right)\left(a^2-d^2\right)=2520\\ \Rightarrow \left(25-4d^2\right)\left(25-d^2\right)=504\\ \Rightarrow 4d^4-125d^2+121=0\\ \Rightarrow \left(d^2-1\right)\left(4d^2-121\right)=0\Rightarrow d=\pm 1,d=\pm \frac{11}{2}\end{array}$

Clearly, $d=\pm 1$ does not give $-\frac{1}{2}$ as a term. So, $d=\pm \frac{11}{2}$

For $d=\frac{11}{R};$ Largest term$=a+2d=5+2\times \frac{11}{2}=16$

For $-d=\frac{11}{R};$ Largest term $=a-2d=5-2\times -\frac{11}{2}=16$