:idea: Maybe Mikey wants to try, but doesn't have enough familiarity with
geometric progressions for a spark to start.
… Which positive real number x has the property that x, ⌊x⌋, and x - ⌊x⌋ form a geometric progression (in that order)? …
This is a neat exercise, for symbolic reasoning. There are different approaches, but I solved the exercise by (1) reasoning from a graph, (2) using the formula for a geometric sequence (progression), (3) solving a quadratic equation and (4) remembering that
a product of reciprocals is 1. The answer is a famous number in geometry!
⌊x⌋ is called the floor function. It leaves Integer values of x alone and rounds non-Integer values down to the next Integer.
To get some sense of the geometric sequence {y₀, y₁, y₂, …}, I began by looking at graphs of:
\(\displaystyle \quad\) y₀ = x
\(\displaystyle \quad\) y₁ = ⌊x⌋
\(\displaystyle \quad\) and their difference
\(\displaystyle \quad\) y₂ = x - ⌊x⌋
The progression is {x, ⌊x⌋, x-⌊x⌋, …}
In other words, for each value of x, the decreasing values y₀,y₁,y₂ lie on a vertical line which intersects the straight graph
x at y₀, the staircase graph
⌊x⌋ at y₁ and (their difference) the saw tooth graph
x-⌊x⌋ at y₂.
The graphs show that x cannot be 1 or less because then
\(\displaystyle \quad\) y₀ =
x
\(\displaystyle \quad\) y₁ = ⌊x⌋ =
0
\(\displaystyle \quad\) y₂ = x - ⌊x⌋ =
x
and {x, 0, x} is not a geometric sequence.
Values of 2 or more don't look promising, either, because the relative distances of [y₀ to y₁] and [y₁ to y₂] don't fit the geometric progression needed (i.e., smaller-smaller-smaller).
The answer x must lie strictly between 1 and 2. Therefore,
what does the graph tell you about the second element in the sequence: y₁? (Think: The graph of y₁ is the staircase.)
We can write a symbolic formula for elements in a geometric sequence like this:
y
n = y₀ ⋅ r
n
where y
n is the nth term of the sequence, y₀ is the first term of the sequence, r is the common ratio and index n starts at zero.
In other words
y₀ = y₀ ⋅ r
0
y₁= y₀ ⋅ r
1
y₂= y₀ ⋅ r
2
Using these symbolic representations for the first three elements of the sequence -- and knowing y₂ is the difference between y₀ and y₁ -- write a quadratic equation and solve it for r.
Once r is known, finding y₀ is easy. (Use the fact printed in green, at the top.)
Okay, Mikey, that's a lot of step-by-step guidance. Have a go at it. Let us know, if you get stuck or there's something I wrote that doesn't make sense (a recurring issue around here). :cool:
has the property that
, and
form a geometric progression (in that order)?
means the greatest integer less than or equal to
