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Question
Find the sum of all even integers between 101 and 999.
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Solution
The even integers between 101 and 999 are:
102, 104...998
Here, we have:
\[a = 102\]
\[d = 2 \]
\[ a_n = 998\]
\[ \Rightarrow 102 + (n - 1)2 = 998\]
\[ \Rightarrow 2n - 2 = 896\]
\[ \Rightarrow 2n = 898\]
\[ \Rightarrow n = 449\]
\[ S_n = \frac{n}{2}\left[ 2a + (n - 1)d \right]\]
\[ \Rightarrow S_{449} = \frac{449}{2}\left[ 2 \times 102 + (449 - 1) \times 2 \right]\]
\[ \Rightarrow S_{449} = \frac{449}{2}\left[ 1100 \right]\]
\[ \Rightarrow S_{449} = 246950\]
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