Slides for 3/22

IB Mathematical Studies SL

Warm-Up Problem

Write the first five terms of the sequence:  a)

a n



n



n

 1 

n

 1  3  b)

a n

 2

a n

 1  7 ;

a

1  8

Guiding Question & Objectives

The early economist Malthus claimed that the world’s food production was, in his time, a large number that was growing in a linear pattern, while the world’s population was a smaller number that was growing in an exponential pattern. Why do you think he was concerned about this? Do you think he was right to be concerned, or have later events proved him wrong?

Today, we will:    Identify

arithmetic

and

geometric

sequences.

Compare and contrast arithmetic and geometric sequences with each other, and with other types of sequences.

Use arithmetic and geometric sequences to make predictions.

Arithmetic Sequences

 An

arithmetic difference

sequence has a common between terms – we add or subtract the same value in each step.

 The recursive formula for an arithmetic sequence always looks like

a n



a n

 1 

d

, where

d

is the common difference.

 The explicit formula for an arithmetic sequence is

a n



a

1 

d



n

 1  .

 Note that this is essentially a

linear

equation!

Example

 In the arithmetic sequence 2, 5, 8, 11, 14, . . .

 What is

d

?

 Write the formula for the sequence in both explicit and recursive form.

Practice

 Find the common difference for the arithmetic sequence 22, 18, 14, 10, . . . and write an explicit formula for the sequence.

Geometric Sequences

  A

geometric

sequence has a common

ratio

between each pair of terms – we multiply or divide by the same value in each step.

The recursive formula for a geometric sequence is

a n



r



a n

 1 , where

r

is the common ratio.

 The explicit formula for a geometric sequence is

a n



a

1 

r



n

 1  .

 Note that this is essentially an

exponential

equation!

Example

 In the geometric sequence 3, 6, 12, 24, 48, . . .

 What is

r

?

 Write both an explicit formula and a recursive formula for the sequence.

Practice

 Find the common ratio for the geometric sequence 1296, 432, 144, 48, . . . and write an explicit formula for the sequence.

Are All Sequences Either Arithmetic Or Geometric?

 No.

Aww.

 The good news is,

most

of the sequences you will encounter will be one of the two.

 The other common types are   factorial sequences, which we will talk about after the IB exam in more detail when we get to combinatorics, and quadratic ones, like square and triangular numbers.

 There are also some other types that are difficult to define explicitly, but easy to define recursively.

 The Fibonacci sequence is one of these.

So How Do We Tell Them Apart?

   If there is a common

difference

between terms, then it’s an arithmetic sequence.

If there’s a common

ratio

between terms, then it’s a geometric sequence.

If it doesn’t have either a common difference or a common ratio, it must be something else!

  If the difference of the differences (the

second difference

) is common to all the terms, then it’s a quadratic of some sort.

Otherwise, look for other patterns.

Example 1

 Identify the type of sequence, and find an explicit formula: 100, 92, 84, 76, 68, . . .

Practice 1

 Identify the type of sequence, and find an explicit formula: 200, 300, 450, 675, . . .

Example 2

 If a sequence is geometric, with

a 2 a

4

=

24 and = 216, write an explicit formula for the sequence.

Practice 2

 If a sequence is geometric, with

a 3 a

6

=

15 and = 72, write an explicit formula for the sequence.

Practice 3

 A staircase is being built out of bricks. Each step uses 2 more bricks than the previous step. If the first step contains 10 bricks, and the staircase is 13 steps, how many bricks are in the last step?

Wrapping Up

 How are arithmetic and geometric sequences similar? How are they different?

 Write a short answer to this question at the end of your notes, then share your answer with your neighbor.

Assignments

 Read Chapter 12 Sections C-D in your textbook.

 Then complete Moodle Assignment 9.2.

 You will have a quiz next class.

 If you have not submitted your IA to TurnItIn.Com yet, you have until midnight tonight.