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.

Sequence	Type of Sequence	Defining Characteristic(s)
1,2,3,4,5,6, …	Arithmetic	Constant difference between successive terms: 1
3,6,9,12,15, …	Arithmetic	Constant difference between successive terms: 3
1,2,4,8,16, …	Geometric	Constant ratio between successive terms: 2
	Geometric	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 4^th 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 8^th 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 n^th term of an arithmetic sequence is given by a_n = a₁ + (n – 1)d. This formula reflects the fact that there are n – 1 common differences between the first and n^th terms of the sequence.

Example 7.7

Find the 81^st 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 n^th term of a geometric sequence with initial term a₁ and common ratio r is given by

a_n = (a₁)(r) ^n-1. For example, the 6^th 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