Section 11.1 Sequences and Summation Notation

Objectives:•Definition and notation of sequences•Recursively defined sequences•Partial sums, including summation notation

Sequences

A sequence is a set of numbers written in a specific order:

a1, a2, a3, a4, …, an, …– The number a1 is called the first term, a2 is the

second term, and in general an is the nth term.

– Since for every natural number n, there is a corresponding number an, we can define a sequence as a function.

Definition of a Sequence

• A sequence is a function f whose domain is the set of natural numbers.

• The values f(1), f(2), f(3), . . . are called the terms of the sequence.

Ex 1. Find the first 5 terms and the 100th term of the sequence defined by each formula.

2

(a) 2 1

(b) 1

n

n

a n

c n

Class Work

Find the first 5 terms and the 100th term of the sequence defined by each formula.

( 1)2)

2

n

n nr

1)

1n

nt

n

Recursively Defined Sequences

Some sequences do not have simple defining formulas like those of the preceding example.

– The nth term of a sequence may depend on some or all of the terms preceding it.

– A sequence defined in this way is called recursive.

Ex 2. Find the first 5 terms of the sequence defined recursively by a1 = 1 and an = 3(an–1 + 2).

Class Work

3. Find the first 6 terms of the sequence defined recursively by a1= 3, a2 = 8 and an = an-1 + an-2.

The Partial Sums of a Sequence

For the sequence a1, a2, a3, a4, …, an, …the partial sums are:

S1 = aS2 = a1 + a2

S3 = a1 + a2 + a3

S4 = a1 + a2 + a3 + a4

Sn = a1 + a2 + a3 + … + an

Ex 3. Find the first four partial sums of the sequence given by . 1

2n

Class Work

4. Find the first four partial sums of the sequence given by . 2 1na n

Sigma Notation

• Given a sequence a1, a2, a3, a4,…we can write the sum of the first n terms using summation notation, or sigma notation.

– This notation derives its name from the Greek letter Σ (capital sigma, corresponding to our S for “sum”).

• Sigma notation is used as follows:

– The left side of this expression is read: “The sum of ak from k = 1 to k = n.”

– The letter k is called the index of summation, or the summation variable.

1 2 3 41

n

k nk

a a a a a a

Ex 4. Find each sum.

5 52

1 3

1(a) (b)

k j

kj

Class Work

Find each sum.

10

5

6

1

5)

6) 2

i

i

i

HW #1 Worksheet Sec 11.1