12

12.1 Define and Use Sequences and Series

Sequence

A function whose domain is a set of consecutive integers

Terms

The values in the range of a sequence

Series

The expression that results when the terms of a sequence are added together

Summation notation

Notation for a series that represents the sum of the terms

Sigma notation

Another name for summation notation, which uses the uppercase Greek letter, sigma, written S

SEQUENCES

A sequence is a function whose domain is a set of _consecutive_ integers. If a domain is not specified,it is understood that the domain starts with 1. The values in the range are called the _terms_ of the sequence.

Domain:1 2 3 4 … n The relative position of each term

Range: a₁ a₂ a₃ a₄… a_n Terms of the sequence

A _finite_ sequence has a limited number of terms.

An _infinite_ sequence continues without stopping.

Finite sequence: 2, 4, 6, 8

Infinite Sequence: 2, 4, 6, 8, …

A sequence can be specified by an equation, or _rule_ .For example, both sequences above can be described by the rule a_n = 2n or f(n) = 2n.

Example 1

Write terms of sequences

Write the first six terms of a_n = 2ⁿ ⁺ ¹.

a₁ = __2^{1 + 1}__ = _4_ 1st term

a₂ = __2^{2 + 1}__ = _8_ 2nd term

a₃ = __2^{3 + 1}__ = _16_ 3rd term

a₄ = __2^{4 + 1}__ = _32_ 4th term

a₅ = __2^{5 + 1}__ = _64_ 5th term

a₆ = __2^{6 + 1}__ = _128_ 6th term

Example 2

Write rules for sequences

Describe the pattern, write the next term, and write a rule for the nth term of the sequence

(a) 1, 4, 9,16, ¼ and (b) 0, 7, 26, 63, …

Solution

a. You can write the terms as _1_², _2_², _3_², _4_². … The next term is
a₅ = _5²_ = _25_. A rule for the nth term is a_n = __n²__.

b. You can write the terms as _1³_ - 1, _2³_ - 1, _3³_ - 1, _4³_ - 1, … The next term is a₅ = _5³_ - 1 = _124_ A rule for the nth term is a_n = _n³ - 1_.

Complete the following exercises.

1. Write the first six terms of the sequence f(n) = 3n - 7.

-4, -1, 2, 5, 8, 11

2. For the sequence -3, 9, -27, 81, …, describe the pattern, write the next term, and write a rule for the nth term.

(-3)¹, (-3)², (-3)³, (-3)⁴; a₅ = - 243; a_n = (-3)ⁿ

SERIES AND SUMMATION NOTATION

When the terms of a sequence are added together, the resulting expression is a series. A series can be finite or infinite.

Finite series: 2 + 4 + 6 + 8

Infinite series: 2 + 4 + 6 + 8 …

You can use __summation__ notation to write a series.

For both series, the index of summation is __i__ and the lower limit of summation is __1__. The upper limit of summation is __4__ for the finite series and __¥__( infinity ) for the infinite series. Summation notation is also called __sigma__ notation because it uses the uppercase Greek letter sigma, written å.

Example 3

Write series using summation notation

Write the series using summation notation.

a. 4 + 7 + 10 + … + 46 b.

Solution

a. Notice that the first term is 3(1) + 1, the second is _3(2) + 1_, the third is _3(3) + 1_, and the last is _3(15) + 1_. So, a_i = _3i+ 1_ where i = 1, 2, 3, …, _15_ The lower limit of summation is _1_ and the upper limit of summation is _15_.

The summation notation for the series is

b. Notice that for each term, the denominator is a perfect cube. So, a_i = where
i = 1, 2, 3, 4 …. The lower limit of summation is _1_ and the upper limit of summation is _infinity_.