7. A, B, C, and D are positive integers. A,B,C forms an arithmetic sequence while B, C, D forms a geometric sequence. if = , what is the smallest possible value of A+B+C+D

Complete QuestionThe positive integers A, B, and C form an arithmetic sequence while the integers B, C, and D form a geometric sequence. If (C/B) = (5/3), what is the smallest possible value of A + B + C + D?Answer:52Step-by-step explanation:If A, B, and C form an arithmetic progressionTheir arithmetic mean, [tex]B=\dfrac{A+C}{2}[/tex]2B=A+CC= 2B-AB, C, D forms a  geometric sequence and Common ratio, r=C/B=5/3The terms in the geometric sequence are:[tex]B, B(\frac{5}{3} ), B(\frac{5}{3} )^2=B, \frac{5B}{3} , \frac{25B}{9}[/tex]Therefore:[tex]C=\frac{5B}{3}\\D= \frac{25B}{9}[/tex]So:[tex]A, B, C, D=A, B, \frac{5B}{3} , \frac{25B}{9}[/tex]From arithmetic sequenceCommon difference,[tex]d=B - A = \frac{5B}{3} - B[/tex][tex]2B -\frac{5B}{3}=A[/tex][tex](2 -\frac{5}{3})B=A\\(\frac{1}{3})B=A\\A=\frac{B}{3}\\[/tex][tex]A, B,C, D =\frac{B}{3},\;B, \;\frac{5B}{3},\;\frac{25B}{9}[/tex]These all have to be positive integers so B must be a multiple of 9, The smallest values are if B is 9A,B,C,D=3,9,15,25So the smallest possible value for: A+B+C+D = 3+9+15+25 = 52