Today in Precalculus

• Notes: Sequences• Homework• Go over quiz

Vocabulary and notation• Sequences: an ordered progression of

numbers.• Term: each number in a sequence is a

termFirst term is a1

Second term is a2

nth term is an

The subscripts denote only the position of the term in the sequence.

Types• Arithmetic Sequence: a sequence in which

there is a common difference, d, between every pair of successive terms.Example: 5,8,11,14

• Geometric: a sequence in which there is a common ratio, r, between every pair of successive terms.Example:

1 1 1 1, , , ,....2 4 8 16

Types• Infinite: there is an infinite number of

terms in the sequenceExample:

• Finite: a finite number of terms in the sequence.Example: 5,8,11,14

• Sequences are infinite unless otherwise specified.

1 1 1 1, , , ,....2 4 8 16

Explicitly Defined Sequence

• A formula is given for any term in the sequenceExample: ak = 2k - 5

Find the first 5 terms and the 20th term for the sequence

a1 = 2(1) – 5 = – 3

a2 = 2(2) – 5 = – 1

a3 = 2(3) – 5 = 1

a4 = 2(4) – 5 = 3

a5 = 2(5) – 5 = 5

a20 = 2(20) – 5 = 35

Recursively Defined Sequence

• The first term is given and along with a rule to obtain each succeeding term from the one preceding it.Example: b1 = 8 and bn = bn-1 – 2 for all n>1

Find the next 4 terms for the sequenceb2 = b1 – 2 = 8 – 2 = 6

b3 = 6 – 2 = 4

b4 = 4 – 2 = 2

b5 = 2 – 2 = 0

General formulas for finding terms in a sequence

• Arithmetic: an = a1 + (n – 1)d

• Geometric: an = a1r(n–1)

• To use these:1) Determine if the sequence is arithmetic or geometric2) Find the common difference or ratio

Example 1• Find the 20th term of the sequence

55,49,43, … and write a recursive and explicit rule.

• Arithmetic sequence with d= -6

• a20 = 55 + (20 – 1)(-6)

a20 = –59

• Recursive rule: ak = ak-1 – 6

• Explicit rule: an = 55 + (n – 1)(-6)

an = 55 – 6n + 6

an = 61 – 6n

Example 2• Find the 8th term of the sequence

and write a recursive and explicit rule.• Geometric sequence with r=4

• Recursive rule: ak = 4ak-1

• Explicit rule:

1 4 16, , ,...5 5 5

8 18

14

5a 8

16,384

5a

114

5n

na 11

(4 ) 45

nna

1(4 )

20n

na

Homework

• Pg 739: 1-9odd, 21-31odd