What is the difference between recursion and induction? I have heard those terms used interchangeably, but I was wondering if there is a difference between them, and if so, what the difference is.
The difference between recursion and induction
definition
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An equation is any expression with an equals sign, so your example is by definition an equation. Equations appear frequently in mathematics because mathematicians love to use equal signs.
A formula is a set of instructions for creating a desired result. Non-mathematical examples include such things as chemical formulas (two H and one O make H2O), or the formula for Coca-Cola (which is just a list of ingredients). You can argue that these examples are not equations, in the sense that hydrogen and oxygen are not "equal" to water, yet you can use them to make water.
Mathematicians have long since realized that when it comes to numbers, certain formulas can be expressed most succinctly as equations. For example, the Pythagorean Theorem $a^2+b^2=c^2$ can be thought of as a formula for finding the length of the side of a right triangle, but it turns out that such a length is always equal to a combination of the other two lengths, so we can express the formula as an equation. The key idea is that the equation captures not just the ingredients of the formula, but also the relationship between the different ingredients.
In your case, "mpg = distance/gallons" is best understood as "a formula in the form of an equation", which means that in this instance the two words are interchangeable.
Let me make an example of something usual in our world, maybe you can find out a sort of differences better. That is the curvature. We are accustomed to think about the flat things on Earth,i.e. the curvature is zero. Making circles, triangles and many many shapes are drew on a piece of a blank paper. Indeed, the curvature of a paper when you put it on a table is $0$. And that is why we learn Ed. Geometry for students on blackboard. This is absolutely OK. But what is the real story? if you want to explain this concept as it is in the real world, you can consider an orange. A surface of an orange (its cover) inside it and outside of it can give us what is the main point about the curvature. Geometrically, when we are working with any spaces with positive curvature,i.e. outside the cover of an orange, if you draw two greater circle, they intersect themselves so, there are no lines parallel to a given line through an outside point (Rejecting the 5-th postulate of Euclidean Geometry).
On the other hand, while in a space of negative curvature, like the surface of an hyperbolic paraboloid or inside the cover an orange , we can draw many lines parallel to a given line through an outside point (Again rejecting the 5-th postulate of Euclidean Geometry). However, when sketching shapes on a blank paper, you are experiencing the zero curvature and inn this case you can easily draw just a single line parallel to a given line through an outside point.
The following figs are from Google's images:
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Best Answer
In my experience:
Extremely often, if a mathematical statement is made about a recursively-defined object, then the proof of that statement will involve induction.
For example, the definition of the Fibonacci numbers is a recursive definition. The proof of the assertion that the $n$th Fibonacci number is at most $2^n$ is an inductive proof.