# Linear Function with Coefficients in Arithmetic Progression

I found an engaging example at Chris Harrow's blog. Chris is the Mathematics Chair at the Hawken School. Chris illustrates the problem in the free online Desmos calculator. I undertook to illustrate it in GeoGebra. Here's the problem:

All linear functions $ax+by+c=0$, where the coefficients $a,b,c$ form an arithmetic sequence, share a certain property. What is it?

The applet below is assumed to be suggestive. There are two sliders: one defines the value of $a$, there other of $d$ such that $b=a+d$ and $c=a+2d$. (For idiosyncratic reasons, GeoGebra display the linear equation as $ax+by=-c.$

If you play with the applet it is pretty hard to miss the point (pun intended): all straight lines that have coefficients in arithmetic progression pass through the point $(1,-2)$. Verification is straightforward:

$a\cdot 1+(a+d)(-2)+(a+2d)=0.$

Chris list several methods by which the students can arrive at this conclusion.

• What Is Line?
• Functions, what are they?
• Cartesian Coordinate System
• Addition and Subtraction of Functions
• Function, Derivative and Integral
• Graph of a Polynomial of arbitrary degree
• Graph of a Polynomial Defined by Its Roots
• Inflection Points of Fourth Degree Polynomials
• Lagrange Interpolation (an Interactive Gizmo)
• Equations of a Straight Line
• Taylor Series Approximation to Cosine
• Taylor Series Approximation to Cosine
• Sine And Cosine Are Continuous Functions