Number & Operations for Teachers 

    Copyright David & Cynthia Thomas, 2009

Arithmetic and Geometric Sequences


Number patterns involving whole numbers, fractions, and decimals are also a common topic in elementary mathematics.  In many cases, the patterns of interest take the form of sequences, or lists.  Table 7.1 presents examples of two types of number sequences: Arithmetic and geometric. 



Type of Sequence

Defining Characteristic(s)

1,2,3,4,5,6, …


Constant difference between successive terms: 1

3,6,9,12,15, …


Constant difference between successive terms: 3

1,2,4,8,16, …


Constant ratio between successive terms: 2



Constant ratio between successive terms: ½

Table 7.1: Sample Sequences


Arithmetic Sequences

Arithmetic sequences are characterized by a common difference between successive terms.  Many children first encounter this sort of sequence as they learn to count-by twos, threes, fives, and so on.  Another natural way to introduce this concept is using a staircase model like that seen in Figure 7.5.  The arithmetic sequence 1, 2, 3, 4, … is modeled in the number of blocks needed to build each step.  The concept of a common difference is modeled in this representation in that each step requires one more block than the step directly to its left.  For young children, the appropriate way to answer the question, “How many blocks are required to build the fifth step” is to actually draw in the next step and count the number of blocks, or to count the number of blocks in the 4th step, then add one.


Figure 7.5: Staircase Model for the Arithmetic Sequence 1, 2, 3, 4, …


How many blocks are needed to build the 8th step?  Older children learn to compute the value of a specified element in an arithmetic sequence based on the first element, a1, of the sequence and the constant difference, d.  For example, the eighth element, a8, in the sequence 3, 6, 9, 12, … may be computed as

a8 = a1 + (8 – 1)d = 3 + (7)(3) = 24


In general, the nth term of an arithmetic sequence is given by an = a1 + (n – 1)d.  This formula reflects the fact that there are n – 1 common differences between the first and nth terms of the sequence.  



Example 7.7

Find the 81st term of the sequence 3,6,9,12,15, … 


Solution 7.7


Geometric Sequences

In geometric sequences, successive terms are related by a common ratio.  For instance, in the geometric sequence 1, 2, 4, 8,16, … , each new term is twice the previous term.  By contrast, each term in the sequence  is ½ the previous term.  This sequence is also geometric.  The nth term of a geometric sequence with initial term a1 and common ratio r is given by

an = (a1)(r) n-1.  For example, the 6th term of the geometric sequence  may be computed as

a6 = (1)(½) 6-1 = (½) 5 = 1/32.  


Like arithmetic sequences, geometric sequences may also be modeled using geometric figures.  Figure 7.6 shows a unit square that is divided into smaller and smaller parts.  The largest division creates two regions, each with area ½ that of the entire square.  Further divisions create smaller and smaller regions whose areas correspond to the terms of the sequence .

Figure 7.6: Area Model for the Geometric Sequence