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.
The second fifth and seventh term of an AP forms first three term of GP find common ratio
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