If the fourth term in an arithmetic sequence is, u4 = 12.5, the tenth is u10 = 27.5. Find the common difference and the 20th term.

The equations for an arithmetic sequences are 1) Un = u1 + (n - 1)d 2) Sn = n/2(2*u1 + (n-1)d) 3) Sn = n/2(u1 + un)

The first step is to calculate the common difference, d. This is done using the first equation Un = u1 + (n - 1)d. We use the fourth term to calculate calculate d:

12.5 = u1 + 3d 27.5 = u1 +9d 15 = 6d d =15/6 d = 2.5

Therefore u1 = 5

For S20 we use the second equation Sn = n/2(2*u1 + (n-1)d).

S20 = 20/2 * (2(5) + (20-1)2.5) = 10 * (10 + 47.5) = 575