+1926 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 $a_1$.
+1946 We can use a clever trick to solve this problem.
We have two equations. Subtracting the first equation from the second equation, we get
(a2−a1)+(a4−a3)+(a6−a5)+(a8−a7)+(a10−a9)+(a12−a11)=0
The reason why this is important is because in every parenthesis, it forms the common difference d.
Thus, we have
6d=0d=0
This means every number in the sequence is the same.
Since the entire sequence is equal to 0, we have a1=0
So our answer is 0.
Thanks! :)
+1946 We can use a clever trick to solve this problem.
We have two equations. Subtracting the first equation from the second equation, we get
(a2−a1)+(a4−a3)+(a6−a5)+(a8−a7)+(a10−a9)+(a12−a11)=0
The reason why this is important is because in every parenthesis, it forms the common difference d.
Thus, we have
6d=0d=0
This means every number in the sequence is the same.
Since the entire sequence is equal to 0, we have a1=0
So our answer is 0.
Thanks! :)
