Find the common difference in an arithmetic seres, where $S_{15} = 322.5$ and the first is $4$ .

1.5

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2.5

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3.5

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4.5

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Solution

The correct option is A $2.5$
$S_{n} = \frac{n}{2} [t_{1} + t_{n}]$
$\Rightarrow 322.5 = \frac{15}{2} [4 + t_{n}]$
$\Rightarrow 322.5 = 7.5 [4 + t_{n}]$
$\Rightarrow 43 = 4 + t_{n}$
$\Rightarrow t_{n} = 39$
$t_{n} = a + (n - 1) d$
$\Rightarrow 39 = 4 + (15 - 1) d$
$\Rightarrow 35 = 14 d$
$\Rightarrow d = 2.5$

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