The third term of an arithmetic sequence is $5$ and the sixth term is $-1.$ Find the twelfth term of this sequence.
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 $12 - 1 = 11.$ That is, the twelfth term is $a + 11d = 9 + 11(-2) = -13.$

