# Straight line intersects geometric sequence

In summary, this approach seems to be able to get all the integers which are not already completely divisible by 2, but it requires a lot of iterations.
Summary:: Two parallel lines (same slope) - one intersects the y-axis, and the other doesn't.
Trying to find the intersection of either with a given geometric sequence.

The lines are:
y=mx
y=mx+1
The values on one or the other of the lines - but not both simultaneously - are to be completely divisible by 2 (end result to be 1). To find out which, I consider an intersection of these with the geometric sequence below.

The geometric sequence:
xn=2n

Is this a workable approach?
Have I made this too murky?

[Moderator's note: moved from a technical forum.]

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What do you mean by one line intersects the y-axis and the other doesn’t and yet they are parallel lines?

Have I made this too murky?

I don't understand what you are trying to do.

jedishrfu said:
What do you mean by one line intersects the y-axis and the other doesn’t and yet they are parallel lines?
Same slope, different lengths.

PeroK said:

I don't understand what you are trying to do.
All positive integers [1→∞) are involved.
Those which are completely divisible by 2 in their original form could conceivably be represented on the mx line (as an idea).
Those which are not at all divisible by 2 must be transformed by adding 1, thus becoming mx+1 (if the above idea holds).
All the numbers which are divisible completely by 2 (both the mx and the mx+1) enter a special domain.
The others must undergo the mx+1 transformation until they become completely divisible by 2.
When the division by 2 of a given number is stopped by some component in the number (e.g., in 6=2⋅3 the component is 3), that component must undergo the transformation.
The process continues until the newly obtained number is completely divisible by 2.

It is evident that the division is assuredly uninterrupted - the only way to enter the new domain - if the divisor is anywhere in the sequence xn=2n

The gist is:
Will all the integers - those which are not already completely divisible by 2 - end up being integers completely divisible by 2 after any number of mx+1 transformations?
If mx and mx+1 are acceptable geometric representations of the classes of integers, I was looking for a way to find their intersections with the sequence mentioned above.

Brute force shows that quintillions of integers do enter the domain after the transformations; sometimes huge number of transformations!
But do all of [1→∞) end up there?!
And how to prove it?

It seems you have an iteration:

2) Take out all factors of ##2## to get ##n_2##, which must be odd.
(If ##n_2 =1 ##, then stop.)
3) Add ##1## to ##n_2## and repeat from step 1.

The question is whether you always reach ##1## at some stage. This is equivalent to getting a power of ##2## at some stage.

It's similar in some ways to the Collatz conjecture:

https://en.wikipedia.org/wiki/Collatz_conjecture

Except, in this case, starting with an odd integer, adding ##1## and dividing by ##2## (or higher powers) will always give a lower number. So, we are bound to get to ##1## eventually.

SammyS and docnet

## 1. How can a straight line intersect a geometric sequence?

A straight line can intersect a geometric sequence if the sequence is graphed on a coordinate plane. The line may intersect with one or more terms of the sequence, depending on the slope and position of the line.

## 2. What does it mean for a straight line to intersect a geometric sequence?

When a straight line intersects a geometric sequence, it means that the line and the sequence share one or more common points on the coordinate plane. These points represent the terms of the sequence that the line passes through.

## 3. Can a straight line intersect a geometric sequence at more than one point?

Yes, a straight line can intersect a geometric sequence at more than one point. This can occur if the line has a slope that is equal to the common ratio of the geometric sequence, resulting in multiple points of intersection.

## 4. How can you determine if a straight line intersects a geometric sequence?

To determine if a straight line intersects a geometric sequence, you can graph both the line and the sequence on a coordinate plane and see if they share any common points. Alternatively, you can calculate the coordinates of the points of intersection using algebraic methods.

## 5. What is the significance of a straight line intersecting a geometric sequence?

The intersection of a straight line and a geometric sequence can provide valuable information about the relationship between the two. It can also be used to solve equations involving both the line and the sequence, or to determine the values of the terms of the sequence at specific points.

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