Question

Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these number is $\dfrac{{ - 1}}{2}$, then the greatest number among them is:

${\text{A}}{\text{. }}\dfrac{{21}}{2} \\\ {\text{B}}{\text{. 27}} \\\ {\text{C}}{\text{. 7}} \\\ {\text{D}}{\text{. 16}} \\\$

Answer 1

Hint:- To solve this question first we have to assume five numbers as in standard format of AP as ( a-2d, a-d, a, a+d, a+2d ) using this standard form of assumption our further calculation becomes easy.

Complete step-by-step answer:
We are given,
Sum of 5 terms of AP is 25. And their product is 2520.
One of them is given $\dfrac{{ - 1}}{2}$
We have to assume five numbers as ( a-2d, a-d, a, a+d, a+2d ) this is the ideal form of assuming five numbers in AP because assuming like this further calculation becomes easy.
The sum of numbers is 25.
a-2d+a-d+a+a+d+a+2d=25
so, a = 5.
Here ‘a’ is third term so third will be 5.
One term of AP is given as $\dfrac{{ - 1}}{2}$.
If $\dfrac{{ - 1}}{2}$ is the first term of AP then, we have to check if it is possible to keep it as the first term of AP.
First term is , a-2d = $\dfrac{{ - 1}}{2}$
$\therefore 2d = 5 + \dfrac{1}{2} \Rightarrow d = \dfrac{{11}}{4}$
Hence if we assume common difference d = $\dfrac{{11}}{4}$ then our AP will be
$\dfrac{{ - 1}}{2},\dfrac{9}{4},5,\dfrac{{31}}{4},\dfrac{{21}}{2}$
Here we can see one number is negative and all are positive that means when we multiply all these we will get a negative number but as given in question multiplication is 2520 which is positive so our assumption is wrong.
So we will assume $\dfrac{{ - 1}}{2}$ as second term of AP
So here second term is a-d= $\dfrac{{ - 1}}{2}$
$a - d = \dfrac{{ - 1}}{2} \Rightarrow d = 5 + \dfrac{1}{2} = \dfrac{{11}}{2}$
So if we assume common difference d = $\dfrac{{11}}{2}$ then our formed AP will be
$- 6,\dfrac{{ - 1}}{2},5,\dfrac{{21}}{2},16$
To verify this assumption is correct or not we multiply and we get 2520 which is exactly the same as given in the question so this assumption is correct.
Hence their greatest number is 16.
So option D is the correct option.

Note: Whenever we get this type of question the key concept of solving is we have to assume five numbers in AP as already written in hint and we have to notice one thing that if we assume numbers as a, a+d, a+2d .... then calculation becomes very lengthy and difficult.