The twenty-first term of an Arithmetic Progression is \(5\frac{1}{2}\) and the sum of the first twenty-one terms is \(94\frac{1}{2}\). Find the : (a) first term ; (b) common difference ; (c) sum of the first thirty terms.
Explanation
(a) \(T_{n} = a + (n - 1)d\) (terms of an AP) \(T_{21} = a + 20d = 5\frac{1}{2}.... (1)\) \(S_{n} = \frac{n}{2} (2a + (n - 1)d) = \frac{n}{2} (a + l)\) Where a and l are the first and last terms respectively. \(S_{21} = \frac{21}{2} (a + 5\frac{1}{2})\)\) \(94\frac{1}{2} = \frac{21}{2} (a + 5\frac{21}{2})\) \(189 = 21 (a + 5\frac{1}{2})\) \(9 = a + 5\frac{1}{2} \implies a = 9 - 5\frac{1}{2} = 3\frac{1}{2}\) (b) Put a in the equation (1), \(3\frac{1}{2} + 20d = 5\frac{1}{2}\) \(20d = 5\frac{1}{2} - 3\frac{1}{2} = 2\) \(d = \frac{2}{20} = \frac{1}{10}\). (c) \(S_{30} = \frac{30}{2} (2(3\frac{1}{2}) + (30 - 1)(0.1)\) = \(15(9.9)\) = \(148.5\)
