If a_m denotes the m^{th} term of an A.P.,the | vedclass

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(C) Let the first term of the $A.P.$ be $A$ and the common difference be $D$.The $m^{th}$ term is given by $a_m = A + (m - 1)D$.The $(m+k)^{th}$ term

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$\frac{a_{m+k} + a_{m-k}}{2}$

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(C) Let the first term of the $A.P.$ be $A$ and the common difference be $D$.The $m^{th}$ term is given by $a_m = A + (m - 1)D$.The $(m+k)^{th}$ term is $a_{m+k} = A + (m + k - 1)D$.The $(m-k)^{th}$ term is $a_{m-k} = A + (m - k - 1)D$.Adding these two terms:$a_{m+k} + a_{m-k} = [A + (m + k - 1)D] + [A + (m - k - 1)D]$$a_{m+k} + a_{m-k} = 2A + (m + k - 1 + m - k - 1)D$$a_{m+k} + a_{m-k} = 2A + (2m - 2)D$$a_{m+k} + a_{m-k} = 2[A + (m - 1)D]$$a_{m+k} + a_{m-k} = 2a_m$Therefore,$a_m = \frac{a_{m+k} + a_{m-k}}{2}$.

If $a_m$ denotes the $m^{th}$ term of an $A.P.$,then $a_m$ =

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