Question

Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.

Answer 1

The odd integers between 1 and 1000 that are divisible by 3 are:
3, 9, 15, 21...999
Here, we have:

\[a = 3, d = 6\]

\[ a_n = 999\]

\[ \Rightarrow 3 + (n - 1)6 = 999\]

\[ \Rightarrow 3 + 6n - 6 = 999\]

\[ \Rightarrow 6n = 1002\]

\[ \Rightarrow n = 167\]

\[ S_n = \frac{n}{2}\left[ 2a + (n - 1)d \right]\]

\[ \Rightarrow S_{167} = \frac{167}{2}\left[ 2 \times 3 + (167 - 1)6 \right]\]

\[ \Rightarrow S_{167} = \frac{167}{2}\left[ 1002 \right] = 83667\]

\[\text { Hence, proved } .\]