Question:
If the 5 th, 7 th and 13 th terms of an AP are in GP, then what is the ratio of its first term to its common difference?
If the 5 th, 7 th and 13 th terms of an AP are in GP, then what is the ratio of its first term to its common difference?
Options:
- a) -3
- b) -2
- c) 2
- d) 3
- a) -3
- b) -2
- c) 2
- d) 3
correct answer : a)
Explanation:
a
5 = a + 4d, a7 = a + 6d and a13 = a + 12d
\ a5, a7 and a13 are in G.P
(a7)2 = a5. a13
(a + 6d)2 = (a + 4d) (a + 12d)
a2 + 12ad + 36d2 = a2 + 16ad + 48d2
12d2 + 4ad = 0 Þ 3d + a = 0
a = -3d Þ a
d = -3
a
5 = a + 4d, a7 = a + 6d and a13 = a + 12d
\ a5, a7 and a13 are in G.P
(a7)2 = a5. a13
(a + 6d)2 = (a + 4d) (a + 12d)
a2 + 12ad + 36d2 = a2 + 16ad + 48d2
12d2 + 4ad = 0 Þ 3d + a = 0
a = -3d Þ a
d = -3