Arithmetic & Geometric Sequences

Just as it is important to be familiar with basic families of functions, it is also important to become acquainted with a few of the most commonly occurring types of sequences. Two of the most common are:

• Arithmetic Sequences: Of the form  a_n = a₀ + r n
  
  
• Geometric Sequences: Of the form  a_n = a₀ r ⁿ

Both types of sequences begin with some pre-designated starting value, a₀ . Subsequent terms are generated either by adding a fixed value of  r  to the previous term (arithmetic), or else by multiplying the previous term by a fixed value of  r  (geometric).

Thus, arithmetic sequences look like this:

a₀ , a₀ + r , a₀ + 2 r , a₀ + 3 r , ...

Geometric sequences look like this:

a₀ , a₀ r , a₀ r ² , a₀ r ³ , ...

For example, the sequence:

7, 3, –1, –5, ...

is arithmetic with  a₀ = 7  and  r = –4 .

The sequence:

7, 3.5, 1.75, 0.875, ...

is geometric with  a₀ = 7  and  r = 1/2 .

Geometric sequences converge to  0  when  |r| < 1  or (trivially) to  a₀  when  r = 1 . They diverge otherwise. (Can you show this?) Arithmetic sequences converge only in the trivial case where  r = 0 . Otherwise they increase or decrease at a constant rate.

Arithmetic and geometric sequences are common in application because they occur naturally in many situations that involve accumulation.

If something is accumulating (or disappearing) at a constant rate, then the quantity will describe an arithmetic sequence over equal intervals of time. Think of water in a reservoir being fed by a constant flow, or snow gently accumulating in a constant flurry.

If something is accumulating (or disappearing) at a rate that is proportional to the amount present, then the quantity will describe a geometric sequence over equal intervals of time. Think of a sequence of filters that each remove a certain proportion of pollutant in a waste stream, or radioactive decay.