Question
The second fifth and seventh term of an AP forms first three term of GP find common ratio
Answers
GPT 3.5
Let the second term of the AP be a, and let the common difference be d.
Then, the second term is (a + d), and the fourth term (which is the second term of the GP) is (a + 3d).
Similarly, the fifth term of the AP is (a + 4d), and the seventh term is (a + 6d).
Since the first three terms of the GP are (a + d), (a + 3d), and (a + 4d), we have:
(a + d) * r = (a + 3d)
(a + 3d) * r = (a + 4d)
Dividing the two equations, we get:
r = [(a + 4d) / (a + 3d)] = (a + 6d) / (a + 4d)
Cross multiplying, we have:
(a + 4d) * (a + 4d) = (a + 3d) * (a + 6d)
Expanding and simplifying:
a^2 + 8ad + 16d^2 = a^2 + 9ad + 18d^2
8ad + 16d^2 = 9ad + 18d^2
d(16d - 8a) = d(18d - 9a)
16d - 8a = 18d - 9a
9a = 2d
a = (2/9) * d
Therefore, the common ratio, r, is:
r = [(a + 4d) / (a + 3d)] = [(2/9) * d + 4d] / [(2/9) * d + 3d]
Simplifying further, we get:
r = [(2/9) + 4] / [(2/9) + 3]
r = (22/9) / (29/9)
r = 22/29
Therefore, the common ratio is 22/29.
Then, the second term is (a + d), and the fourth term (which is the second term of the GP) is (a + 3d).
Similarly, the fifth term of the AP is (a + 4d), and the seventh term is (a + 6d).
Since the first three terms of the GP are (a + d), (a + 3d), and (a + 4d), we have:
(a + d) * r = (a + 3d)
(a + 3d) * r = (a + 4d)
Dividing the two equations, we get:
r = [(a + 4d) / (a + 3d)] = (a + 6d) / (a + 4d)
Cross multiplying, we have:
(a + 4d) * (a + 4d) = (a + 3d) * (a + 6d)
Expanding and simplifying:
a^2 + 8ad + 16d^2 = a^2 + 9ad + 18d^2
8ad + 16d^2 = 9ad + 18d^2
d(16d - 8a) = d(18d - 9a)
16d - 8a = 18d - 9a
9a = 2d
a = (2/9) * d
Therefore, the common ratio, r, is:
r = [(a + 4d) / (a + 3d)] = [(2/9) * d + 4d] / [(2/9) * d + 3d]
Simplifying further, we get:
r = [(2/9) + 4] / [(2/9) + 3]
r = (22/9) / (29/9)
r = 22/29
Therefore, the common ratio is 22/29.