AJC Prelims 1999
An arithmetic sequence has first term a and common difference d. It is given that the sum of the first four terms is less than the sum of the next four terms by 8. Also, the first, third and sixth term of the sequence are three consecutive terms of a geometric progression. Find the exact values of a and d.
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Answer:
Given first term a and common difference d
S4 < S8 - S4
(4/2) (2a + 3d) < (8/2) (2a + 7d) - (4/2) (2a + 3d)
4 (2a + 3d) < 4 (2a + 7d)
4d > 0
d > 0 ------- (1)
T1, T3, T6 are in G.P.
(T3)2 = T1 * T6
(a + 2d)2 = (a)(a + 5d)
4d2 - ad = 0
d (4d - a) = 0
d = 0 (reject since d > 0) or d = a/4
Choose a = 4 and d = 1
