jmc

Question

If a,b,c,d are positive real numbers such that a,b,c,d form an increasing arithmetic sequence and a,b,d form a geometric sequence, then find the value of ad.

Accepted Answer

We have b=a + r, c=a + 2r, and d=a + 3r, where r is a positive real number. Also, b^2 = ad yields (a+r)^2 = a(a+3r), or r^2=ar. It follows that r=a and d = a + 3a = 4a. Hence ad = 14.