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Question
Find the sum of odd integers from 1 to 2001.
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Solution
\[\text { The odd integers from 1 to 2001 are }1, 3, 5 . . . . . 2001 . \]
\[\text { It is an AP with a = 1 and d = 2 } . \]
\[ a_n = 2001\]
\[ \Rightarrow 1 + (n - 1)2 = 2001\]
\[ \Rightarrow 2n - 2 = 2000\]
\[ \Rightarrow 2n = 2002\]
\[ \Rightarrow n = 1001\]
\[\text { Also }, S_{1001} = \frac{1001}{2}\left[ 2 \times 1 + \left( 1001 - 1 \right)2 \right]\]
\[ \Rightarrow S_{1001} = \frac{1001}{2}\left[ 2 \times 1 + \left( 1000 \right)2 \right]\]
\[ \Rightarrow S_{1001} = \frac{1001}{2} \times 2002 = 1002001\]
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