Arithmetic and Geometric Progressions

Progressions are sequences whose terms move in a controlled, increasing or decreasing pattern.

Arithmetic Progression (AP). A sequence of the form

is an AP: each term (after the first) is obtained by adding a fixed constant $d$ — the common difference — to the term before it; $a$ is the initial (first) term. The $n^{th}$ term is

$\sqrt2,\sqrt2+\sqrt3,\sqrt2+2\sqrt3,\ldots$ is an AP with common difference $\sqrt3$; $12,9,6,3,\ldots$ is an AP with common difference $-3$. It is a pleasant fact that $3,7,11$ (three primes) form an AP, and more generally $T_n=an+b$ with $\gcd(a,b)=1$ produces an AP containing infinitely many primes along with infinitely many composites.

Geometric Progression (GP). A sequence of the form

is a GP: each term (after the first) is obtained by multiplying the term before it by a fixed constant $r$ — the common ratio. The $n^{th}$ term is

$1,2,4,8,16,\ldots$ is a GP with common ratio $2$; $\sqrt2,2,2\sqrt2,4,4\sqrt2,\ldots$ is a GP with common ratio $\sqrt2$. Taking logarithms of a GP with positive common ratio produces an AP: if $a,ar,ar^2,\ldots$ is a GP with $r>0$, then $\log a,\log(ar),\log(ar^2),\ldots$ is an AP with common difference $\log r$.

The constant sequence $c,c,c,\ldots$ is always an AP ($d=0$); it is also a GP ($r=1$) provided $c\ne0$. …