In an A.P, $a_7=9$  and if $a_1{a}_2{a}_7$  is least then common difference is

 $given that a_7=a+6d=9,a_1{a}_2{a}_7=a(a+d)(a+6d)$ 

$=\left(9-6d\right) \left(\left[9-6d\right]+d\right)\left(9\right)=$$\left(9-6d\right)\left(9-5d\right)9$  $=27\left(3-2d\right)\left(9-5d\right)$

  $=27\left(10d^2-33d+27\right)$ $=270\left(d^2-\frac{33}{10}d+\frac{27}{10}\right) =270\left({\left(d-\frac{33}{20}\right)}^2-\frac{1089}{400}+\frac{27}{10}\right)$

$=270\left\{{\left(d-\frac{33}{20}\right)}^2-\frac{9}{400}\right\}$  is least when $d=\frac{33}{20}$