The twenty-first term of an Arithmetic Progression is (5frac{1}{2}) and the sum of the first twenty-one terms is (94frac{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 = 5frac{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 + 5frac{1}{2})))
(94frac{1}{2} = frac{21}{2} (a + 5frac{21}{2}))
(189 = 21 (a + 5frac{1}{2}))
(9 = a + 5frac{1}{2} implies a = 9 – 5frac{1}{2} = 3frac{1}{2})
(b) Put a in the equation (1),
(3frac{1}{2} + 20d = 5frac{1}{2})
(20d = 5frac{1}{2} – 3frac{1}{2} = 2)
(d = frac{2}{20} = frac{1}{10}).
(c) (S_{30} = frac{30}{2} (2(3frac{1}{2}) + (30 – 1)(0.1))
= (15(9.9))
= (148.5)