A three-digit even number is given such that the hundreds, tens, and unit digits respectively form a decreasing arithmetic sequence. When two is added to the number, the same order of the digits now forms a geometric sequence. What is the original number?
+2401 100a + 10b + c
a, b, c is a decreasing arithmetic sequence.
a, b, c + 2 is a geometric sequence.
Note that the c can't carry over since that only happens when c is 9 (odd number) or 8, but 8 + 2 = 0 which can't be part of a geometric sequence.
c = c
b = c + x
a = c + 2x
b^2 = (c + 2)(a)
(c + x)^2 = (c + 2)(c + 2x)
c^2 + 2cx + x^2 = c^2 + 2cx + 2c + 4x
x^2 = 2c + 4x
Let's try when c = 2.
x^2 = 4 + 4x, x is a weird number.
Let's try when c = 4.
x^2 = 8 + 4x, x is not a nice number.
Let's try when c = 6.
x^2 = 12 + 4x, x is a nice number.
(x-2)(x-6) = 0
x = 6, this is not possible.
Let's try when c = 0.
x^2 = 4x
x = 4.
c = 0
b = 4
a = 8
Our number is 840. :))
=^._.^=
