Four distinct positive integers form an arithmetic sequence and a geometric sequence.
Arithmetic and Geometric
Four distinct positive integers (A1, A2, A3, and A4) form an arithmetic sequence. A1, A2, and A4 form a geometric sequence. The sum of the four terms is a perfect cube. What is the smallest possible value of A1?
Problem Moderated by: Douglas
Problem Solution
Call the terms a-2d, a-d, a, and a+d.
Now, (a+d)/(a-d)=(a-d)/(a-2d), or a=3d. The terms of the sequence, then, are d, 2d, 3d, and 4d, which sums to (2)(5)d, which must be a perfect cube - therefore, the value of the first term, d must be (2^2)(5^2)=100.
