The second fifth and seventh term of an AP forms first three term of GP find common ratio
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.
Let's denote the second term of the arithmetic progression (AP) as a, and the common difference as d.
The second term in the AP is a + d.
The fifth term in the AP is a + 4d.
The seventh term in the AP is a + 6d.
Since the first three terms of the arithmetic progression also form a geometric progression (GP), we can write the relationship:
(a + d) / a = (a + 4d) / (a + d) = (a + 6d) / (a + 4d)
To find the common ratio, let's solve this equation:
(a + d) / a = (a + 4d) / (a + d)
Cross-multiplying:
(a + d)(a + d) = a(a + 4d)
a² + ad + ad + d² = a² + 4ad
Simplifying and canceling:
a² + 2ad + d² - a² - 4ad = 0
-2ad + d² = 0
Factoring out d:
d(-2a + d) = 0
Therefore, either d = 0 or -2a + d = 0.
If d = 0, then the terms of the AP would be equal, and it wouldn't actually form an AP.
So, we can conclude that -2a + d = 0.
Solving for d:
-2a + d = 0
d = 2a
Hence, the common ratio in the GP formed by the first three terms of the AP is 2a.