Arithmetic and Geometric Progressions

by Dave
(DAVE)—

In this video, I am going to explain what arithmetic and geometric progressions are. I’ll also give several basics examples. Then I will prove the summations formulas for both the arithmetic and geometric progressions. In the end, I’ll discuss how these progressions arise in nature.

I discuss the similarities between arithmetic and geometric progressions. After that, I give examples of each type and explain the recursive versions of each one. In the following, I also discuss both types of sequences, and for each one, I prove the summation formula.

Arithmetic progressions

An arithmetic progression, sometimes called an arithmetic sequence, is a sequence of numbers with a fixed common difference. For example, the sequence $10, 20, 30, 40, …$ is an arithmetic sequence with the common difference of 10. 

Definition. A sequence of the form $$a, a+d, a+2d, \ldots, a+nd, \ldots$$ where $a,d\in \mathbb{R}$ is called an arithmetic progression.

In addition, an arithmetic progression can be written as a recursive formula as $$a_0 = a, \qquad a_n = a_{n-1} + d \qquad n\geq 1.$$

Theorem. Let $a,d\in \mathbb{R}.$ Prove that for every positive integer $n$, that $$a+(a+d)+(a+2d)+\cdots + (a+nd) =\frac{(2a+nd)(n+1)}{2}.$$

Proof 1. A proof using mathematical induction.

Using mathematical induction to evaluate the summation formula of an arithmetic progression
Using mathematical induction to evaluate the summation formula of an arithmetic progression

Proof 2.

The summation formula of an arithmetic progression
The summation formula of an arithmetic progression

Geometric Progressions

geometric progression, sometimes called a geometric sequence, is a sequence of numbers with a fixed common ratio. For example, the sequence 10, 100, 1000, 10000, … is an geometric sequence with the common ratio of 10. 

Definition. A sequence of the form $$a, ar, ar^2, \ldots, a r^n, \ldots$$ where $a,r\in \mathbb{R}$ and $r\neq 1$ is called a geometric progression.

In addition, a geometric progression can be written as a recursive formula as $$a_0 = a, \qquad a_n = a_{n-1} r \qquad n\geq 1.$$

Theorem. Let $a,r\in \mathbb{R}$ and suppose $r\neq 1.$ Prove that, each positive integer $n$, that $$a+ar +ar^2+\cdots +a r^n = a\left(\frac{r^{n+1}-1}{r-1}\right).$$

Proof 1. A proof using mathematical induction.

Using mathematical induction to evaluate the summation formula of a geometric progression
Using mathematical induction to evaluate the summation formula of a geometric progression

Conclusion

So when are the arithmetic and geometric progressions used in every day life?

Whenever you are counting something, you are using an arithmetic progression. For example, when $a=0$ and $d=1$, the arithmetic progression is just $0,1,2,3, \ldots.$ So whether you are counting by twos, threes, or any fixed number, you are using arithmetic progressions. For example, velocity. On the other hand, if you have exponential growth (or decay) you are using a geometric progression. For example, when $a=1$ and $r=2$, the geometric progression is just $1,2,4,8,16,\ldots.$ For example, acceleration.

To learn a great deal more on this topic, consider taking the online course The Natural Numbers.