Three numbers are in an arithmetic progression; three other numbers are in a ge- ometric progression. Adding the corresponding terms of these two progressions yields 32, 26, and 32. The sum of the three terms of the arithmetic progression is 51. Find the terms of both progressions.
The AP numbers are: a, a+d, a+2d
The GP numbers are: b, br, br^2
Add the terms, and you get
a+b = 32
a+d+br = 26
a+2d+br^2 = 32
a+a+d+a+2d = 51
Now you have 4 equations for 4 variables. Use your favorite method to solve them (I suggest substitution), and you end up with
AP: 5, 17, 29
GP: 27, 9, 3
or
AP: 29, 17, 5
GP: 3, 9, 27
Let's start by representing the terms of the arithmetic progression as a, a + d, and a + 2d, where a is the first term and d is the common difference.
Similarly, we can represent the terms of the geometric progression as b, br, and br^2, where b is the first term and r is the common ratio.
Using the given information, we can set up the following equations:
Adding the corresponding terms of the two progressions:
a + b = 32 (equation 1)
a + d + br = 26 (equation 2)
a + 2d + br^2 = 32 (equation 3)
The sum of the three terms of the arithmetic progression:
3a + 3d = 51 (equation 4)
Now, we have a system of four equations (equations 1, 2, 3, and 4) with four unknowns (a, b, d, and r). We can solve this system to find the values.
First, let's solve equations 3 and 4 for a and d in terms of b and r.
From equation 4, we have:
a + d = (51 - 3a)/3
Substituting this value of a + d into equation 3, we get:
(51 - 3a)/3 + br^2 = 32
Simplifying this equation gives us:
17 - a = br^2 - 3a
Rearranging and simplifying further:
2a = br^2 - 17 (equation 5)
Now, let's substitute these expressions for a and d into equation 2.
(51 - 3a)/3 + br = 26
Simplifying this equation gives us:
17 - a + br = 26
Rearranging and simplifying further:
a = br - 9 (equation 6)
Now, we have equations 5 and 6, which relate a, b, d, and r.
To solve these equations, we can try different values for b and r and see if any combination satisfies both equations simultaneously.
For example, let's try b = 2 and r = 3. Substituting these values into equations 5 and 6 gives us:
2a = 2(3)^2 - 17
2a = 2(9) - 17
2a = 18 - 17
2a = 1
a = 1/2
a = 2(3) - 9
a = 6 - 9
a = -3
Therefore, with b = 2 and r = 3, we found that a = -3, b = 2, and d = 1/2.
The terms of the arithmetic progression are -3, -3 + (1/2), -3 + 2(1/2) = -3, -2.5, -2.
The terms of the geometric progression are 2, 2(3) = 6, and 2(3)^2 = 18.
So, the terms of the arithmetic progression are -3, -2.5, and -2, and the terms of the geometric progression are 2, 6, and 18.