### Instruction: The third term of an arithmetic sequence is $5$ and the sixth term is $-1.$ Find the twelfth term of this sequence.### Response: To find the twelfth term of an arithmetic sequence, I need to know the first term and the common difference.
I can use the information given to find both of these.
Let $a$ be the first term and $d$ be the common difference.
Then the third term is $a + 2d = 5$ and the sixth term is $a + 5d = -1.$
I can solve this system of equations by subtracting the first equation from the second, which gives me $3d = -6,$ or $d = -2.$
Then I can plug this value of $d$ into either equation to find $a.$ For example, using the first equation, I get $a + 2(-2) = 5,$ or $a = 9.$
Now I have $a = 9$ and $d = -2.$
To find the twelfth term, I need to add $11$ times the common difference to the first term, since $12 - 1 = 11.$