What is an "arithmetic progression"?

Question

Today, while doing my math problems on polynomials, I came across the phrase "arithmetic progression." Despite trying hard, I just can't find out the meaning of this phrase and cannot solve my problems.

Any help or hint would be much appreciated!

Answer 1

Think of the name itself. First, by "progression", it obviously means a sequence of numbers. Now comes the question "What type of sequence?"

Now, in mathematics, especially in sequence and series, the term "arithmetical growth" means that the growth is constant between terms, i.e., the difference between two consecutive terms are the same".

And... there's your answer. An arithmetic progression is a sequence of values where the difference between two consecutive terms is the same (called common difference and usually denoted by $d$). Here's an example:

$$\{1,3,5,7,9,11\}$$

You can see that any two consecutive terms have a difference of $2$. So, this is an arithmetic progression with first term $1$, last term $11$ and a common difference of $2$.

Same goes when it comes to decreasing arithmetic progression. It isn't necessary that the progression be increasing. For example,

$$\{-1,-2,-3,-4,-5,-6,-7\}$$

We can see that the terms decrease by $1$. So, this is an arithmetic progression with first term $-1$, last term $-7$ and a common difference of $-1$.

This is the basic intuitive definition. For the formal definition, you can look it up on Wikipedia.