9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32,...

9-4 Sequences & Series

Basic Sequences

Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2k, … {1/k: k = 1, 2, 3, …} (a1, a2, a3, …, ak, …}

Finite Sequence: has a definite end / last term Infinite Sequence: continues infinitely

Explicit vs. Recursive

• Explicit formula: A function used to find the required term.

• Recursive formula: A function that uses the previous terms to find the required term.

Explicit Sequence

• Ex: Find the first 6 terms and the 100th term of the explicitly-defined sequence

cn = n3 – n

c1

c2

c3

c4

c5

c6

c100

Recursive Sequence

• Ex: Find the first 4 terms and the 8th term of the recursively-defined sequence

a1 = 8 and an = an-1 – 4, for n ≥ 2

a1

a2

a3

a4

a8

Arithmetic SequenceThe pattern is addition!

• A sequence {an} is an arithmetic sequence if it can be written explicitly in the form

an = a1 + (n – 1)d

for some constant d, where d is the common difference (aka pattern number)

• Each term can be obtained recursively by

an = an-1 + d (for all n ≥ 2)

Arithmetic Sequence Example

• Ex: For the arithmetic sequence below, finda) The common difference

b) The tenth term

c) A recursive rule for the nth term

d) An explicit rule for the nth term

6, 10, 14, 18, …

You try!

• Ex: For the arithmetic sequence below, finda) The common difference

b) The tenth term

c) A recursive rule for the nth term

d) An explicit rule for the nth term

4, 1, -2, -5, …

Geometric SequenceThe pattern is multiplication!

• A sequence {an} is a geometric sequence if it can be written explicitly in the form

an = a1 · r n – 1

for some nonzero constant r, where r is the common ratio (aka pattern number)

• Each term can be obtained recursively by

an = an-1 · r (for all n ≥ 2)

Geometric Sequence Example

• Ex: For the geometric sequence below, finda) The common ratio

b) The tenth term

c) A recursive rule for the nth term

d) An explicit rule for the nth term

2, 6, 18, 54, …

You try!

• Ex: For the geometric sequence below, finda) The common ratio

b) The tenth term

c) A recursive rule for the nth term

d) An explicit rule for the nth term

1, -2, 4, -8, 16, …

Constructing Sequences

• Ex: The second and fifth terms of a sequence are 6 and 48, respectively. Find explicit and recursive formulas for the sequence if it is a) arithmetic and b) geometric.

Fibonacci Sequence

• A famous example of a recursive sequence• a1=0, a2=1, an = an-1 + an-2 for n ≥ 3

• 0, 1, 1, 2, 3, 5, 8, 13, 21, …• Named for Leonardo of Pisa• Appears everywhere in nature (check

phyllotaxy in Biology)• Be amazed by Fibonacci

Super Doodle Girl!• http://youtube.com/v/P0tLbl5LrJ8

It’s a race!

• Who can be the first one to find the sum of all numbers from 1 – 100 ?

Sigma Notation• This is a shorthand way to represent a large

sum of numbers• Uses the capital Greek letter sigma, Σ• In summation notation, the sum of the terms of

the sequence {a1, a2, …, an} is denoted

which is read “the sum of ak from k=1 to n”

• The variable k is called the index of summation

…Say what?!!??

• See if you can determine the number represented by each of the following expressions:

1. 2. 3.

Sum of a Finite Arithmetic Sequence

• Let {a1, a2, a3, …, an} be a finite arithmetic sequence with common difference d. Then the sum of the terms of the sequence is

• Proof is on pg 740 if you’re in the mood for some fun!

Revisit Arithmetic Sequences• Remember our example 3, 6, 9, 12, 15?

Find the sum for this sequence. Use the formula.

• What about the sum of numbers 1 – 100?

Sum of a Finite Geometric Sequence

• Let {a1, a2, a3, …, an} be a finite geometric sequence with common ratio r ≠1. Then the sum of the terms of the sequence is

S =

• Proof is on pg 742 if you want more fun!

Revisit Geometric Sequences• Remember our example 2, 4, 8, 16, 32?

Find the sum for this sequence. Use the formula.

• Find the sum for 42, 7, , …,

Infinite Series:• Used when adding an infinite number of

terms together• Not a true sum; how can you find an answer

for infinity?• We use a sequence of partial sums and

limits to find these infinite sums• We can only find the sums if the series

converges to a single value. If it diverges, the limit DNE and we have no sum.

Does it converge?

• For each of the following series, find the first five terms in the sequence of partial sums. Which of the series appear to converge?

1. 0.3 + 0.03 + 0.003 + 0.0003 + …

2. 1 – 2 + 3 – 4 + 5 – 6 + …

Sum of an Infinite Geometric Series

• The geometric series converges

if and only if |r| < 1. If it does converge,

the sum is S =

• Try this formula with #1 from the last slide!

One more neat trick…

• Ex: Express the repeating decimal 7.1414141414 in fraction form.