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?
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?
arithmetic progression
A
B,
C= A+2(B-A)
geometric progression
B r=C/B=5/3
B, B(5/3), B(5/3)^2
B, 5B/3, 25B/9
so
C=5B/3
D=25B/9
A, B, 5B/3, 25B/9
B - A = 5B/3 - B
2B-5B/3 = A
(2-5/3)B = A
A=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
\( \frac{9}{3},\;9, \;\frac{45}{3},\;\frac{9*25}{9}\\ 3,\;9, \;15,\;25\\ \)
So the smallest possible value for A+B+C+D = 3+9+15+25 = 52
AP = 1, 3, 5...............etc.
GP = 3, 5, 8 1/3.........etc.
A + B + C + D =1 + 3 + 5 + 8 1/3 = 17 1/3
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?
arithmetic progression
A
B,
C= A+2(B-A)
geometric progression
B r=C/B=5/3
B, B(5/3), B(5/3)^2
B, 5B/3, 25B/9
so
C=5B/3
D=25B/9
A, B, 5B/3, 25B/9
B - A = 5B/3 - B
2B-5B/3 = A
(2-5/3)B = A
A=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
\( \frac{9}{3},\;9, \;\frac{45}{3},\;\frac{9*25}{9}\\ 3,\;9, \;15,\;25\\ \)
So the smallest possible value for A+B+C+D = 3+9+15+25 = 52