+834 Let $a_1,$ $a_2,$ $a_3,$ $\dots,$ $a_{10},$ $a_{11},$ $a_{12}$ be an arithmetic sequence. If $a_1 + a_3 + a_5 + a_7 + a_9 + a_{11} = 0$ and $a_2 + a_4 + a_6 + a_8 + a_{10} + a_{12} = 0$, then find $a_1$.
+129847 We can write
a1 + (a1 + 2d) + (a1 + 4d) + (a1 + 6d) + (a1 + 8d) + (a1 + 10d) = 6a1 + 30d = 0
(a1 + d) + (a1 + 3d) + (a1 + 5d) + (a1+ 7d)+ (a1 + 9d) + (a1 + 11d) = 6a1 + 36d = 0
So
6a1 + 30d = 0
6a1 + 36d = 0
6a1 + 30d = 0
-6a1 - 36d = 0 add these
-6d = 0
d = 0
So
6a1 + 30(0) = 0
6a1 = 0
a1 = 0
