Question

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

• A. $16$
• B. $27$
• C. $7$
• D. $\frac{21}{2}$

Answer 1

The correct option is A

$16$

Explanation for correct option:

Step1. Finding middle term of Arithmetic progression.

Let the five terms of Arithmetic progression is $(a-2d),(a-d),a,(a+d),(a+2d)$ and the difference between the terms is $d$

Given, sum of the terms is $=25$

Therefore,

$\begin{array}{rcl}(a-2d)+(a-d)+a+(a+d)+(a+2d)& =& 25\\ 5a& =& 25\\ a& =& 5\end{array}$

Step2. Finding common difference of the term.

Product of the terms is $2560$

Therefore,

$\begin{array}{rcl}(a-2d)\times (a-d)\times a\times (a+d)\times (a+2d)& =& 2520\\ (a^2-4d^2)\times (a^2-d^2)a& =& 2520\\ (25-4d^2)\times (25-d^2)5& =& 2520\\ (25-4d^2)\times (25-d^2)& =& 504\\ 4d^4-125d^2+625& =& 504\\ 4d^4-125d^2+121& =& 0\\ 4d^4-4d^2-121d^2+121& =& 0\\ 4d^2(d^2-1)-121(d^2-1)& =& 0\\ (d^2-1)(4d^2-121)& =& 0\\ d& =& 1or\frac{11}{2}\end{array}$

Step3. Finding greatest term of Arithmetic progression.

Now $a+2d$ will be the greatest terms of the given Arithmetic Progression

Therefore, $5+2\times \frac{11}{2}=16$

Hence, correct option is $\mathbf{\left(}\mathit{A}\mathbf{\right)}$