1st, 7th and 25th terms of an AP are in three consecutive terms of GP. The 20th term of an AP is 22. Find

Question

Common difference of an AP
The 10th of the GP

Answer 1

Let's call the first term of the AP "a" and the common difference of the AP "d".
Given that the 1st, 7th, and 25th terms of the AP are in three consecutive terms of a GP, we can write the following equations:

a + (6d) = ar
a + (24d) = ar^2

Simplifying these equations, we have:
1) ar - a = 6d
2) ar^2 - a = 24d

Since the 20th term of the AP is 22, we can write another equation:
a + (19d) = 22

From equation 1), we have:
ar = 6d + a
r = (6d + a)/a

Substituting r in equation 2), we have:
(6d + a)r^2 - a = 24d
(6d + a)((6d + a)^2/a^2) - a = 24d
(6d + a)(36d^2 + 12ad + a^2)/a^2 - a = 24d

Expanding and simplifying further, we have:
216d^3/a^2 + 96d^2 + 42d + a = 24d

Rearranging the terms, we have:
216d^3/a^2 + 96d^2 + 18d - 22d + a = 0

Since the 20th term of the AP is 22, we also have:
a + 19d = 22

We have a system of equations here that we can solve to find the values of a and d. However, finding the common difference (10th term of the GP) is not possible with the provided information.