Sequences  April 26, 2016

PART ONE !

Warm Up1.) 26, 21, 16, ___, ___, ___, ___

2.) O, T, T, F, F, S, S, E, ____

3.) J, F, M, A, M, J, ____

Find the pattern and fill in the blanks!

Sequences  April 26, 2016

Fill in the table values and graph the function at the top of the handout.

When connected, the points create a _________________, which means f(x) is a ____________________ function.

Sequences  April 26, 2016

Fill in the table values and graph the following discrete function ­ DO NOT CONNECT THE DOTS!! The notation will look awkward at first. We ask that you treat n like x, and that you treat a like y.

Did you connect the points? What does leaving the points unconnected mean?

What does it mean that there are no points to the left of (1 , ­2)?

Sequences  April 26, 2016

The equation you graphed is called a discrete function. 

Let's look at the a­values that we came up with.

When we look at a list of numbers like this, they are collectively called a sequence. Note that this sequence has a very a very specific starting value (­2), but no specified ending value. This is common.

Note also that the points you graphed are linear, and that the terms (numbers) in the sequence are all separated by the same additive value (2).  This makes it an arithmetic sequence.  

Sequences  April 26, 2016

Find the next numbers in the pattern.

4, 9, 14, 19, ____, ____, ____

­8, ­2, 4, 10, ____, ____, ____

Arithmetic Sequences

Sequence: set of numbers

Term: each number in a sequence

Arithmetic Sequence: each term after the first is found by adding a constant to the previous term

Common Difference (d): constant you add to get next term

***found by subtracting any term by its previous term

Sequences  April 26, 2016

an = a1 + (n - 1)dan = term in the sequence

a1 = first term in the sequence

n = # of terms in the sequence

d = common difference

If the first term of the arithmetic sequence is 7 and the common difference equals 3...what is the 279th term?

Do you REALLY want to write out all 279 terms???

Sequences  April 26, 2016

Write a rule for the nth term of the arithmetic sequence. Then find a25.

1.) 6, 14, 22, 30, 38 ...

2.) ‐7, ‐3, 1, 5, 9 ...

Write a rule for the arithmetic sequence given ...

3.) a5 = 50 Common Difference = 0.25

4.) a20 = ‐111 Common Difference = ‐6

Sequences  April 26, 2016

PART TWO !https://www.youtube.com/watch?v=kkGeOWYOFoA

Warm Up

1.) 4, 16, 64, ___, ___, ___, ___

2.) 32, 16, 8, ___, ___, ___, ___

Sequences  April 26, 2016

Fill in the table values and graph the following discrete function 

What continuous function does it resemble?

Let's look at the a­values that we came up with.

Again, when we look at a list of numbers like this, they are collectively called a sequence. Note that this sequence has a very a very specific starting value (1), but no specified ending value. Again, this is common.

Sequences  April 26, 2016

Note also that the points you graphed are non­linear, and that the terms (numbers) in the sequence are all not separated by the same additive value. This makes it a geometric sequence.  

Example 1: Example 2:

4, 8, 16, 32, _______, _______, _______  ­2, 6, ­18, 54, _______, _______, _______

Sequences  April 26, 2016

Geometric Sequences

Geometric Sequence: each term after the first is found by multiplying a constant to the previous term.

Common Ratio (r): constant you multiply to get next term.

(found by dividing any term by its previous term)

an = a1 r n ­ 1

an = term in the sequence

a1 = first term in the sequence

n = # of terms in the sequence

r = common ratio

GEOMETRIC

SEQUENCE

Sequences  April 26, 2016

1.) Write the first six terms of a geometric sequence given:

a1 = 4 & r = 3

____, ____, ____, ____, ____, ____

2.) Write the first six terms of a geometric sequence given:

a1 = 125 & r =

____, ____, ____, ____, ____, ____

25

-

3.) a4 = 10 r = n = 1212

Wrtie the rule and find the nth term for the given geometric sequence.

Sequences  April 26, 2016

Write a rule for the nth term of the geometric sequence. Then find a15

4.) 5, 10, 20, 40, ...

a.) Write the rule. b.) Find a15

5.) 6, -30, 150, -750, ...

a.) Write the rule. b.) Find a15