What is the greatest three-digit number "abc'' such that \(4, a, b\) forms a geometric sequence and \(b, c, 5\) forms an arithmetic sequence?
c-b=5-c
2c+b=5
a*a/4=b
2c+a^2 /4=5
we try to make a as big as possible while still being an interger
a = 4
c is not interger
a=2 is the biggest
so c=2
and b=1
+128408 Here's my best attempt....
4, a, b b, c, 5
b + (c - b) + (c - b) = 5
2c - b = 5
b = 2c - 5
4b = 8c - 20 (1)
4 * (a/4)^2 = b
a^2/4 = b
a^2 = 4b (2) equate (1) and (2)
a^2 = 8c - 20
c a b
9 x
8 x
7 6 9
6 x
5 x
4 x
3 2 1
The largest possible "abc" is 697
Proof
4, 6 , 9 is geometrc if r = 1.5
9, 7, 5 is arithmetic if d = -2
