Warm Up 1. First: state whether each series below is arithmetic or not. 2. Then, find the sum of...

Warm Up

1. First: state whether each series below is arithmetic or not.

2. Then, find the sum of each series:a.

Warm Up #2

Mrs. Inscoe is watching the leaves fall in her yard. On the first day, she sees 5 leaves fall. The next day, she sees 15, and on the third day she sees 25. How many leaves will she have to rake up if she knows that the trees in her yard will lose leaves for 30 days?

Homework/Classwork Questions?

Geometric Series

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QUIZ TOMORROW!Covers:

All sequences (Arithmetic, Geometric) All series (Arithmetic, Geometric, neither)

Test NEXT WEDNESDAY, NOVEMBER 7th

A geometric sequence is…A list of numbers that are all multiplied by

the same thing to get the next number.A geometric series is…The SUM of all of the numbers in a geometric

sequence.

The sum of a geometric series with n terms or the nth partial sum of a geometric series can be found with one of the two formulas below:

But what if I don’t know n?

This formula is useful when:

Formula of a Finite Geometric Series

)1(

)1(1r

raS

n

n

This formula is useful when:

r

raaS nn

11

Find the sum of the first six terms of the geometric series 8 + 14 + 24.5 + …

First: Find the common ratio.

Then: Plug what you know into the appropriate formula to find the sum of the series.

Example:

Find the sum of the first n terms of a geometric series with a1 = 3, an = 768, and r = -2

Decide which formula you should use.

Plug your numbers into the formula and simplify!

You try two.

Find the sum of the first 11 terms of the geometric series 7 + (-24.5) + 85.75 + …

Find the sum of the first n terms of a geometric series with a1 = -8, an = 131,072, and r = -4

Sigma Notation

We can also represent geometric series using sigma notation.

The good news: The sigma notation works the same way that it did yesterday.

This time, we just use our geometric series formula instead of our arithmetic one!

Example

Find

7

2

1)5(3n

n You can use either formula for this one! Which one would you like to use?

You try these two:

31

16

1)2(5.0n

n

11

4

1)5.0(120n

n

How about this one?

What’s wrong with this one?

4

1)2.0(4n

n

Dealing with Infinity

Can we find the sum of an infinite group of numbers?

Sometimes.If the numbers get closer and closer to

zero as we keep going in our sequence, then we can figure out approximately what it would equal if you added them together forever.

Dealing with Infinity

Remember this example from yesterday?

We found the seventh partial sum and kept adding 3’s

What fraction does this decimal approach?

n

na

10

13

So how do we tell if the numbers in a sequence approach zero?

Arithmetic series NEVER approach zero.For geometric series, the only way for the

numbers to get smaller and smaller (aka approach zero), is if we are. multiplying them by a number smaller than one

What do we call that number that we multiply the terms by each time? And what variable represents it?

Common Ratio/r

Our numbers in a geometric sequence approach zero if:

|r|<1

Can I find the infinite sum of the following geometric sequences? (Is |r|<1?)

Finding the Sum of Infinite Geometric Series

Finite Geometric Series:

As n ∞, rn 0

Infinite Geometric Series:

)1(

)1( 01

r

raSn

)1(1

r

aSn

)1(1

r

aSn

Examples: Find the sum of each infinite geometric series, if possible

9 + 3 + 1 + … 0.25 + (-1.25) + 6.25 + …

ALWAYS FIND r FIRST!

How about the one from earlier?

4

1)2.0(4n

n

You try these on your own!

Find the sum of the infinite series below if possible:

10 + (-5) + 2.5 + … 20 + (-30) + 45 + …

1

1)8.0(120n

n

Real World Example

A new restaurant opens and serves 20 customers on its opening night. Of the 20 customers, 16 found the experience enjoyable and each told 2 friends over the next month. This group each told 3 friends of the next month, and so on, for 5 months. Assuming that no one heard twice, how many people have had a positive experience or heard positive reviews of the restaurant?

Complete the following problems by yourself or with the people on your row.

Finishing ____ problems before the end of class will earn you some Smarties!

Homework:

Finish class workSTUDY FOR YOUR QUIZ