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Question
Find the sum of all odd numbers between 100 and 200.
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Solution
All the odd numbers between 100 and 200 are:
101, 103...199
Here, we have:
\[a = 101\]
\[d = 2\]
\[ a_n = 199\]
\[ \Rightarrow 101 + (n - 1) \times 2 = 199\]
\[ \Rightarrow 2n - 2 = 98\]
\[ \Rightarrow 2n = 100\]
\[ \Rightarrow n = 50\]
\[ S_n = \frac{n}{2}\left[ 2a + (n - 1)d \right]\]
\[ \Rightarrow S_{50} = \frac{50}{2}\left[ 2 \times 101 + (50 - 1)2 \right]\]
\[ \Rightarrow S_{50} = 25\left[ 202 + 98 \right]\]
\[\Rightarrow S_{50} = 7500\]
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