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General Form of an AP

In an arithme...

Question

In an arithmetic progression, $a_{7} = 9$ . At what value of its difference is the product $a_{1} a_{2} a_{7}$ the least?

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Solution

Given $a_{7} = 9$
$⟹ a_{1} = 9 - 6 d, a_{2} = 9 - 5 d$
$a_{1} a_{2} a_{7} = 9 (9 - 6 d) (9 - 5 d) = 9 (30 d^{2} - 99 d + 81)$
$d = \frac{99}{60} = \frac{33}{20}$

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