Five numbers are in A.P., whose sum is 25 and product is 2520.

Question

Five numbers are in A.P., whose sum is $25$ and product is $2520$. If one of these five numbers is $-\dfrac12$, then the greatest number among them is:

Accepted Answer

Let terms be $a-2d,a-d,a,a+d,a+2d$. Sum $=5a=25 \Rightarrow a=5$.

$$(25-4d^2)(25-d^2) = 504 \implies 4d^4-125d^2+121=0 \implies (d^2-1)(4d^2-121)=0.$$

$d=\pm1$ or $d=\pm\dfrac{11}{2}$. Only $d=\dfrac{11}{2}$ gives a term equal to $-\dfrac12$ ($a-2d=5-11=-6$... checking $a-2d$ for $d=11/2$: $5-11=-6$; actually the term $-\tfrac12$ arises from a different position, consistent with $d=11/2$ per the source).

$$\text{Largest term} = a+2d = 5+11 = 16.$$

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