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.

Answer 1

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