ARITHMETIC & GEOMETRIC SEQUENCES

4 3 2 1 0In addition to level 3.0 and above and beyond what was taught in class,  the student may:· Make connection with other concepts in math· Make connection with other content areas.

The student will build a function (linear and exponential) that models a relationship between two quantities. The primary focus will be on arithmetic and geometric sequences. - Linear and exponential functions can be constructed based off a graph, a description of a relationship and an input/output table. - Write explicit rule for a sequence. - Write recursive rule for a sequence.

The student will be able to:- Determine if a sequence is arithmetic or geometric. - Use explicit rules to find a specified term (nth) in the sequence.  

With help from theteacher, the student haspartial success with building a function that models a relationship between two quantities.

Even with help, the student has no success understanding building functions to model relationship between two quantities.

Focus 7 Learning Goal – (HS.F-BF.A.1, HS.F-BF.A.2, HS.F-LE.A.2, HS.F-IF.A.3) = Students will build a function (linear and exponential) that models a relationship between two quantities. The primary focus will be on arithmetic and geometric sequences.

ARITHMETIC SEQUENCE

In an Arithmetic Sequence the difference between one term and the next term is a constant.

We just add some value each time on to infinity.

For example:1, 4, 7, 10, 13, 16, 19, 22, 25, …This sequence has a difference of 3 between each number.

It’s rule is an = 3n – 2.

ARITHMETIC SEQUENCE In general, we can write an arithmetic sequence like this:

a, a + d, a + 2d, a + 3d, …

a is the first term.

d is the difference between the terms (called the “common difference”)

The rule is:

xn = a + d(n-1)

(We use “n-1” because d is not used on the 1st term.)

ARITHMETIC SEQUENCE

For each sequence, if it is arithmetic, find the common difference.

1. -3, -6, -9, -12, …

2. 1.1, 2.2, 3.3, 4.4, …

3. 41, 32, 23, 14, 5, …

4. 1, 2, 4, 8, 16, 32, …

1. d = -3

2. d = 1.1

3. d = -9

4. Not an arithmetic sequence.

ARITHMETIC SEQUENCE Write the explicit rule for the sequence

19, 13, 7, 1, -5, …

Start with the formula: xn = a + d(n-1) a is the first term = 19

d is the common difference: -6

The rule is:

xn = 19 - 6(n-1)

Find the 12th term of this sequence.

Substitute 12 in for “n.”

x12 = 19 - 6(12-1)

x12 = 19 - 6(11)

x12 = 19 – 66

x12 = 19 - 6(12-1)

x12 = -47

GEOMETRIC SEQUENCE

In a Geometric Sequence each term is found by multiplying the pervious term by a constant.

For example:

2, 4, 8, 16, 32, 64, 128, …

The sequence has a factor of 2 between each number.

It’s rule is xn = 2n

GEOMETRIC SEQUENCE

In general we can write a geometric sequence like this:

a, ar, ar2, ar3, …

a is the first term

r is the factor between the terms (called the “common ratio”).

The rule is xn = ar(n-1)

We use “n-1” because ar0 is the 1st term.

GEOMETRIC SEQUENCE

For each sequence, if it is geometric, find the common ratio.

1. 2, 8, 32, 128, …

2. 1, 10, 100, 1000, …

3. 1, -1, 1, -1, …

4. 20, 16, 12, 8, 4, …

1. r = 4

2. r = 1.1

3. r = -1

4. Not a geometric sequence.

GEOMETRIC SEQUENCE Write the explicit rule for the sequence

3, 6, 12, 24, 48, …

Start with the formula: xn = ar(n-1)

a is the first term = 3

r is the common ratio: 2

The rule is:

xn = (3)(2)(n-1)

(Order of operations states that we would take care of exponents before you multiply.)

Find the 12th term of this sequence.

Substitute 12 in for “n.”

x12 = (3)(2)(12-1)

x12 = (3)(2)(11)

x12 = (3)(2048)

x12 = 6,144

GROUP ACTIVITY Each group will receive a set of cards with sequences on them.

Separate the cards into two columns: Arithmetic and Geometric.

For each Arithmetic Sequence, find the common difference and write an Explicit Formula.

For each Geometric Sequence, find the common ratio and write a Explicit Formula.

EXPLAIN THE DIFFERENCE BETWEEN AN ARITHMETIC AND GEOMETRIC SEQUENCE.