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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 $a_1$.

 May 24, 2024
 #1
avatar+129489 
+1

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

 

cool cool cool

 May 24, 2024

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