Find the 9th term of the arithmetic sequence.
a7=-42 a11=-28

Question

Find the 9th term of the arithmetic sequence. 
a7=-42 a11=-28

imstuck · Answer

The first term, a1 is 7.  The formula for an arithmetic sequence is $$a _{n}=a _{1}+(n-1)d$$.  You need to write to equations and solve them simultaneously for a1 and d, then rewrite the equations.  Lets see how this works out. The first equation is -42 = a1 + (7-1) d; the second one is -28 = a1 + (11-1) d.  We could subtract the 1 from the 7 and that's 6, and then 11-1 = 10.  Now we have -42 = a1 + 6d and
                                        -28 = a1 + 10d.
Multiply the first equation by -1 to rid yourself of the a1.  Now we have
                                        42 = -a1 - 6d and
                                      -28 =  a1 + 10d.  The a1's cancel out, leaving
                    
                                       14 = 4d.  Divide by 4 and get d = 3.5.  Now fill in the 3.5 in one of the equations to find a1.  -42 = a1 + 6(3.5).  That gives us -63 is a1, which makes sense, since the 7th term is -42 and the 11th term is -28.  Got that?