First slide

Arithmetic progression

Question

In an A.P, $a_{7} = 9$ and if $a_{1} a_{2} a_{7}$ is least then common difference is

Difficult

A

$\frac{11}{30}$
B

$\frac{13}{10}$
C

$\frac{32}{33}$
D

$\frac{33}{20}$

Solution

$g i v e n t h a t a_{7} = a + 6 d = 9, a_{1} a_{2} a_{7} = a (a + d) (a + 6 d)$
$= (9 - 6 d) ([9 - 6 d] + d) (9) =$ $(9 - 6 d) (9 - 5 d) 9$ $= 27 (3 - 2 d) (9 - 5 d)$
$= 27 (10 d^{2} - 33 d + 27)$ $= 270 (d^{2} - \frac{33}{10} d + \frac{27}{10}) = 270 ({(d - \frac{33}{20})}^{2} - \frac{1089}{400} + \frac{27}{10})$
$= 270 {{(d - \frac{33}{20})}^{2} - \frac{9}{400}}$ is least when $d = \frac{33}{20}$

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