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  4. An A.P. consists of n (odd terms) and its middle term is m. Then the sum of the A.P. is

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Q. An A.P. consists of n (odd terms) and its middle term is m. Then the sum of the A.P. is

Sequences and Series

Solution:

Middle term $ \frac{T_{n+1}}{2} = a+\left(\frac{n+1}{2} -1\right)d = m$
$\Rightarrow 2a+\left(n+1-2\right)d = 2m $
$\Rightarrow 2a+\left(n-1\right)d = 2m $
$ \Rightarrow \frac{n}{2}\left[2a+\left(n-1\right)d\right] = \frac{n}{2}\left(2m\right) = mn $
$\Rightarrow S_{n}$ of $A.P. = mn$