Question

7. A, B, C, and D are positive integers. A,B,C forms an arithmetic sequence while B, C, D forms a

Answer 1

Complete Question

The 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:

52

Step-by-step explanation:

If A, B, and C form an arithmetic progression

Their arithmetic mean, $B=\dfrac{A+C}{2}$

2B=A+C

C= 2B-A

B, C, D forms a  geometric sequence and Common ratio, r=C/B=5/3

The terms in the geometric sequence are:

$B, B(\frac{5}{3} ), B(\frac{5}{3} )^2=B, \frac{5B}{3} , \frac{25B}{9}$

Therefore:

$C=\frac{5B}{3}\\D= \frac{25B}{9}$

So:

$A, B, C, D=A, B, \frac{5B}{3} , \frac{25B}{9}$

From arithmetic sequence

Common difference,$d=B - A = \frac{5B}{3} - B$

$2B -\frac{5B}{3}=A$

$(2 -\frac{5}{3})B=A\\(\frac{1}{3})B=A\\A=\frac{B}{3}$

$A, B,C, D =\frac{B}{3},\;B, \;\frac{5B}{3},\;\frac{25B}{9}$

These all have to be positive integers so B must be a multiple of 9, The smallest values are if B is 9

A,B,C,D=3,9,15,25

So the smallest possible value for:

A+B+C+D = 3+9+15+25 = 52