If the common difference of the $A . P$ . in which $T_{7} = 9$ and $T_{1}, T_{2}, T_{7}$ is least, is $^{'} d^{'}$ then $20 d$ is-

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Solution

$T_{1} = a$

$T_{2} = a + d$

$T_{7} = a + 6 d = 9 - 6 d$

$T_{1} T_{2} T_{7} = a (a + d) (a + 6 d) = N$

$N = (9 - 6 d) (9 - 5 d) 9$

$N = (6 d - 9) (5 d - 9) 9$

$\frac{d N}{d d} = 0$

$d = \frac{33}{20}$

then $20 d = \frac{33}{20} \times 20$

$= 33$

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