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Question
Find the sum of the following serie:
101 + 99 + 97 + ... + 47
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Solution
101 + 99 + 97 + ... + 47
Here, the series is an A.P. where we have the following:
\[a = 101\]
\[d = \left( 99 - 101 \right) = - 2\]
\[ a_n = 47\]
\[ \Rightarrow 101 + (n - 1)( - 2) = 47\]
\[ \Rightarrow 101 - 2n + 2 = 47\]
\[ \Rightarrow 2n - 2 = 54\]
\[ \Rightarrow 2n = 56\]
\[ \Rightarrow n = 28\]
\[ S_n = \frac{n}{2}\left[ 2a + (n - 1)d \right]\]
\[ \Rightarrow S_{28} = \frac{28}{2}\left[ 2 \times 101 + \left( 28 - 1 \right) \times ( - 2) \right]\]
\[ = \frac{28}{2}\left[ 2 \times 101 + 27 \times ( - 2) \right] \]
\[ = 2072\]
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