Question

Let $a_i,i=1,2,\dots ,n$ be an A.P. If $a_7=9,$ then the value of common difference of the A.P. such that $a_1\cdot a_2\cdot a_7$ is minimum, is

• A. $\frac{11}{20}$
• B. $\frac{33}{20}$
• C. $2$
• D. $3$

Answer 1

The correct option is B $\frac{33}{20}$

$a_7=a_1+6d$
$a_2=a_1+d$
$a_1\cdot a_2\cdot a_7=a_1(a_1+d)9=9a_1(a_1+\frac{9-a_1}{6})$
$=\frac{3}{2}(5a_1^2+9a_1)$

Let $f\left(a_1\right)=\frac{3}{2}(5a_1^2+9a_1)$
$f^{\prime }\left(a_1\right)=\frac{3}{2}(10a_1+9)$
$\therefore a_1=-\frac{9}{10}$ is the point of minima of $f$.
$\therefore d=\frac{9+\frac{9}{10}}{6}=\frac{33}{20}$