The first term of an arithmetic series is a and the common difference is d. The 12th term is 66.5 and the 19th term is 98. Write down two equations in a and d then solve these simultaneous equations to find a and d.

The first step is to recall the formula for arithmetic progressions: u(n) = a + (n-1)d We can then put all the information given in the question into this so u(12) = 66.5 = a + 11d and u(19) = 98 = a + 18d By lining up the two simultaneous equations as below, we can see if we take the first equation away from the second the a terms will cancel out: 66.5 = a +11d 98 = a + 18d By taking away the first from the second we get 31.5 = 7d from which we find d = 4.5 We can then use this value in the first equation because we now know 66.5 = a + 114.5 = a + 49.5 By rearranging we find a = 17 Using the second equation we can check both these values are right and we are done!