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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 find the greatest number amongst them.
Given: Five numbers are in A.P., whose sum is $25$ and product is $2520$. If one of these five numbers is $-\frac{1}{2}$.
To do: To find the greatest number amongst them.
Solution:
Let us assume the term to be: $a-2d,\ a-d,\ a,\ a+d,\ a+2d$.
As given sum of these terms is $25$.
$\therefore a-2d+a-d+a+a+d+a+2d=25$
$\Rightarrow 5a=25$
$\Rightarrow a=5$
​
Also product $=2520$
$\Rightarrow( a-2d)( a+2d)( a-d)( a+d).a=2520$
$( a^2-4d^2)( a^2-d^2)a=2520$
$( 25-4d^2)( 25-d^2)=504$
$4d^4-125d^2+121=0$
$( d^2-1)( 4d^2-121)=0$
If $d^2-1=0$
$\Rightarrow d^2=1$
$\Rightarrow d=\pm 1$ it will not give $-\frac{1}{2}$ value of any terms.
​
or
If $4d^2-121=0$
$\Rightarrow 4d^2=121$
$\Rightarrow d^2=\frac{121}{4}$
$\Rightarrow d=\pm \frac{11}{2}$
​
So we will consider $d=±\frac{11}{2}$
​
So largest term is $5+2\times\frac{11}{2}=16$
 
Updated on: 10-Oct-2022
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