Find four numbers that form a geometric progression such that the second term is less than the first by 36 and the third term is greater than the fourth term by 324.
(Hint: There are two possible sets of four numbers)
To find the set of four numbers that form a geometric progression, we can start by assuming the first term as 'a' and the common ratio as 'r'.
Let's denote the four numbers in the geometric progression as:
a, ar, ar^2, ar^3
Given that the second term is less than the first by 36, we have:
ar = a - 36 ----(1)
Also, the third term is greater than the fourth term by 324, so we have:
ar^2 - ar^3 = ar^3 - ar^4 + 324
ar^2 - ar^3 - ar^3 + ar^4 = 324
ar^2(1 - r) - ar^3(1 - r) = 324
ar^2 - ar^3 = 324/(1 - r) ----(2)
From equation (1), we can substitute ar as a - 36 in equation (2):
(a - 36) - (a - 36)r = 324/(1 - r)
a - 36 - ar + 36r = 324/(1 - r)
a(1 - r) + 36r = 324/(1 - r)
a(1 - r)^2 = 324/(1 - r)
a(1 - 2r + r^2) = 324/(1 - r)
a - 2ar + ar^2 = 324/(1 - r)
a(1 - 2r + ar) = 324/(1 - r)
Since a cannot be zero for a non-zero geometric progression, we can divide both sides by a:
1 - 2r + ar = 324/(a * (1 - r))
At this point, we can try different values for 'a' and 'r' to see if any satisfy the equation. For example, let's assume 'a' as 1, and 'r' as 2:
1 - 2(2) + 1(2^2) = 324/(1 * (1 - 2))
1 - 4 + 4 = 324/-1
1 = -324 (Not true)
We can try different values for 'a' and 'r' until we find a set of numbers that satisfies the equation. Once we find one solution, we can use the formula ar^3 to find the fourth term. And we can use the formula ar^(-1) to find the second term.
Keep in mind that there are two possible sets of numbers in a geometric progression that satisfy the given conditions.