Question

If `m-1, 3m-2, 5m` is a geometric sequence, then what is the value of `m` ?

Answer 1

There is no Real value of `m` resulting in a geometric sequence.

It is possible to get a geometric sequence of Complex numbers with:

`m = 7/8+-sqrt(15)/8i`

Explanation:

If `a, b, c` is a geometric sequence, then `b/a = c/b` and hence `b^2 = ac`.

So in order for `m-1, 3m-2, 5m` to be a geometric sequence, we must have:

`(3m-2)^2 = (m-1)(5m)`

which expands to:

`9m^2-12m+4 = 5m^2-5m`

Subtract `5m^2-5m` from both sides to get:

`4m^2-7m+4 = 0`

The discriminant `Delta` of a quadratic `ax^2+bx+c` is given by the formula:

`Delta = b^2-4ac`

So in the case of this quadratic in `m` (which has `a=4`, `b=-7`, `c=4`), we find:

`Delta = (-7)^2-4(4)(4) = 49-64 = -15`

Since `Delta < 0` there are no Real zeros. We can find Complex zeros using the quadratic formula:

`m = (-b+-sqrt(b^2-4ac))/(2a)`

`= (-b+-sqrt(Delta))/(2a)`

`= (7+-sqrt(15)i)/8`

`= 7/8+-sqrt(15)/8i`

These values lead to geometric sequences of Complex numbers.