If a, b, c, and 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 a/d is

Question

If a, b, c, and 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 a/d is 
A) 1/12 B) 1/6 C) 1/4 D) 1/3 E) 1/2

GPT 3.5 · Answer

Let the common difference of the arithmetic sequence be $d$. Then $b=a+d$ and $d=b+d$, so $3d=b$ and $d=b/3$. Also, since $a$, $b$, $d$ form a geometric sequence, we know that $(a+d)/a=a/d=b/a$. Substituting $d=b/3$, we get $(a+b/3)/a=a/(b/3)=3a/b$. Therefore, $a/d=3a/b=\boxed{	extbf{(D)}\ 1/3}$.