An arithmetic progression has a tenth term (a10) = 11.1 and a fiftieth term (a50) = 7.1 Find the first term (a) and the common difference (d). Also find the sum of the first fifty terms (Sn50) of the progression.

We start off by constructing simultaneous equations as there are two variables - a and d - that we do not know. We use the formula:a_n = a + (n-1)di) 11.1 = a + 9dii) 7.1 = a + 49d
i) - ii) gives you 4 = - 40d
Rearrange to make d the subject, you get d = 4/-40 = -0.1To find a, we plug in d to one of our equations and rearrange to get a.So using i) we get a = 11.1 - 9d which is a = 11.1 + 0.9 = 12
Therefore, a = 12 and d= -0.1
To get the sum of the first 50 terms, we use the formula S_n = n/2 (2a + (n-1)d)So S₅₀= 25 (24 + (49 x -0.1)) = 477.5Therefore the sum of the first 50 terms is 477.5.