**Arithmetic and Geometric Sequences**

The sequence 5, 7, 9, 11, 13,… , 5+2(*n*-1), … , where each term after the first is obtained by adding 2 to the preceding term, is an Example of an arithmetic sequence. The sequence 5, 10, 20, 40, 80,… , 5⋅2^{(n-1)}, … , Where each term after the first is obtained by multiplying the preceding term by 2, is an Example of a geometric sequence.

**Determine whether each sequence is arithmetic, geometric, or neither. Explain.**

1). 200, 40, 8, … .

Solution:

Since the ratios are constant, the sequence is geometric. The common ratio is ⅕.

2). 2, 4, 16, … .

Solution:

The ratios are not constant, so the sequence is not geometric.

There is no common difference, so the sequence is not arithmetic.

Thus, the sequence is neither geometric nor arithmetic.

3). -6, -3, 0, 3, … .

Solution:

The ratios are not constant, so the sequence is not geometric.

Since the differences are constant, the sequence is arithmetic. The common difference is 3.

4). 1,-1,1,-1,…

Solution:

Since the ratios are constant, the sequence is geometric. The common ratio is -1.

5). 4, 1, 2, … .

Solution:

Find the ratios of consecutive terms.

The ratios are not constant, so the sequence is not geometric.

Find the ratios of the differences of consecutive terms

There is no common difference, so the sequence is not arithmetic.

Thus, the sequence is neither geometric nor arithmetic.

6). 10, 20, 30, 40, … .

Solution:

Find the ratios of consecutive terms.

The ratios are not constant, so the sequence is not geometric.

Find the differences of consecutive terms.

Since the differences are constant, the sequence is arithmetic. The common difference is 10.

7). 4, 20, 100, … .

Solution:

Find the ratios of consecutive terms.

Since the ratios are constant, the sequence is geometric. The common ratio is 5.

8). 212, 106, 53, … .

Solution:

Find the ratios of consecutive terms.

Since the ratios are constant, the sequence is geometric. The common ratio is ½.

9). -10, -8, -6, -4, … .

Solution:

Find the ratios of consecutive terms.

The ratios are not constant, so the sequence is not geometric.

Find the differences of consecutive terms.

Since the differences are constant, the sequence is arithmetic. The common difference is 2.

10). 5, -10, 20, 40, … .

Solution:

Find the ratios of consecutive terms.

The ratios are not constant, so the sequence is not geometric.

Find the differences of consecutive terms.

There is no common difference, so the sequence is not arithmetic. Thus, the sequence is neither geometric nor arithmetic.

Two kinds of regular sequences occur so often that they have specific names, **arithmetic** and **geometric sequences**. We treat them together because some obvi- ous parallels between these kinds of sequences lead to similar formulas. This also makes it easier to learn and work with the formulas. The greatest value in this association is understanding how the ideas are related and how to derive the formulas from fundamental concepts. Anyone learning the formulas this way can recover them whenever needed.

Both arithmetic and geometric sequences begin with an arbitrary first term, and the sequences are generated by regularly adding the same number (the **common difference** in an arithmetic sequence) or multiplying by the same number (the **common ratio** in a geometric sequence). Definitions emphasize the parallel features, which examples will clarify.

Definition: arithmetic and geometric sequences

Arithmetic Sequencea_{1}=aanda=_{n}a_{(n-1)}+d forn>1

The sequence {a} is an arithmetic sequence with strong>first term_{n}aandcommon differenced.

Geometric Sequencea_{1}=aanda=r⋅_{n}a_{(n-1)}forn>1

The sequence {a} is a geometric sequence with_{n}first termaandcommon ratior.

The definitions imply convenient formulas for the *n*th term of both kinds of sequences. For an arithmetic sequence we get the *n*th term by adding d to the first term n-1 times; for a geometric sequence, we multiply the first term by r, n-1 times.

Formulae for the

nth terms of arithmetic and geometric sequences

For an arithmetic sequence, a formula for thenth term of the sequence isa=a+(_{n}n-1)d. (1)

For a geometric sequence, a formula for thenth term of the sequence isa=_{n}a⋅r^{(n-1)}. (2)

The definitions allow us to recognize both arithmetic and geometric sequences. In an arithmetic sequence the difference between successive terms, *a*_{(n+1)}–*a _{n}*, is always the same, the constant

*d*; in a geometric sequence the ratio of successive terms, , is always the same.

Given the structure of arithmetic and geometric sequences, any two terms completely determine the sequence. Using Equation (1) or (2), two terms of the sequence give us a pair of equations from which we can find the first term and either the common difference or common ratio, as illustrated in the next example.

Example 1: Arithmetic or geometric?

The first three terms of a sequence are given. Determine if the sequence could be arithmetic or geometric. If it is an arithmetic sequence, find *d*; for a geometric sequence, find *r*.

(a) 2, 4, 8, … (b) ln 2, ln 4, ln 8,… (c) ½,⅓,¼,…

Solution

Strategy: Calculate the dif- ferences and/or ratios of successive terms.

(a) *a*_{2}–*a*_{1}=4-2=2, and *a*_{3}–*a*_{2}=8-4=4. Since the differences are not the same, the sequence cannot be arithmetic. Checking ratios, , and , so the sequence could be geometric, with a common ratio *r*=2. Without a formula for the general term, we cannot say anything more about the sequence.

(b) *a*_{2}–*a*_{1}=ln 4-ln 2=ln(4/2)=ln 2, and *a*_{3}–*a*_{2}=ln 8-ln 4=ln(8/4)=ln 2, so the sequence could be arithmetic, with ln 2 as the common difference. As in part (a), we cannot say more because no general term is given.

(c) *a*_{2}–*a*_{1}=⅓-½=-⅙, and *a*_{3}–*a*_{2}=¼-⅓=-1/12. The differences are not the same, so the sequence is not arithmetic. , and , so the sequence is not geometric. Note that the sequence in part (a) could be geometric and the sequence in part (b) could be arithmetic, but in part (c) you can conclude unequivocally that the sequence cannot be either arithmetic or geometric.

Example 2: Arithmetic or geometric?

Determine whether the sequence is arithmetic, geometric, or neither.

(a) {3-1.6n} (b) {2^{n}} (c) *a _{n}*=ln

*n*

solution:

(a)

*a*

_{2}–

*a*

_{1}=(3-1.6⋅2)-(3-1.6⋅1)=(-0.2)-1.4=-1.6, and

*a*

_{3}–

*a*

_{2}=(3-1.6⋅3)-(3-1.6⋅2)=-1.6. From the first three terms, this could be an arithmetic sequence with

*d*=-1.6. Check the difference

*a*

_{(n+1)}–

*a*.

_{n}*a*

_{(n+1)}–

*a*=[3-1.6(

_{n}*n*+1)]-[3-1.6

*n*]=-1.6

The sequence is arithmetic, with

*d*=-1.6.

(b)

*a*

_{2}–

*a*

_{1}=4-2=2, and

*a*

_{3}–

*a*

_{2}=8-4=4, so the sequence is not arithmetic. Using the formula for the general term,

The sequence {2

^{n}} is geometric, with 2 as the common ratio.

(c)

*a*

_{(n+1)}–

*a*=ln(

_{n}*n*+1)-ln

*n*=ln[(

*n*+1)/

*n*]. The difference depends on n, so the sequence is not arithmetic. Checking ratios, , so the ratio also changes with n. The sequence is neither arithmetic nor geometric.