Algebra 2, with Trig
Algebra 2 12.1 – 12.4 Sequences & Series
12.1-12.3 Arithmetic and Geometric Sequences
__________________ - an ordered collection of numbers. It’s a function whose domain is a set of consecutive integers. The values in the range are called the terms of the sequence. A sequence can either be finite or infinite.
Finite sequence 2, 4, 6, 8 Infinite sequence 2, 4, 6, 8, …
(IT ENDS!) (DOESN’T END)
Write the 1st 5 terms of an = 2n + 3.
a1 = 2(1) + 3
a2 = 2(2) + 3
a3 =
a4 =
a5 =
If there is a pattern to the sequence, you may be able to write a rule for the nth term.
Ex. 1 ([pic]) = ([pic], [pic], [pic], [pic]
The next term is _________ So, an = ______________
Ex. 2 Choose the general equation. (2, 6, 10, 14, …)
a) y = 3x - 1
b) y = 2x + 4 (Hint: Use the table of values on calculator)
c) y = x + 2
d) y = 4x – 2
Arithmetic sequence – difference between consecutive terms is a constant.
General Rule for an Arithmetic Sequence: [pic],
where …
a1 is the __________ term
d is the _____________________
n is the _____________________
an is the value of the _________ term
Decide whether each sequence is Arithmetic, or neither. If it is Arithmetic, find [pic] using [pic].
1. [pic], [pic], [pic], [pic]. . . A or n 2. −4, 4, 12, 20, 28 . . . A or n
Geometric sequence – there’s a common ratio between terms.
General Rule for a Geometric Sequence: [pic]
where
a1 is the _________ term of the sequence
r is the ______________________
n is the ______________________
an is the _________ term of the sequence
Ex. 1, 2, 4, 8, 16, 32, … What is the ratio (r) ? _______
Find [pic]using [pic].
Decide whether each sequence is Arithmetic, Geometric, or neither. If it is Arithmetic or Geometric, find [pic]by using either [pic] or [pic].
3. 1, 2, 6, 24, 120, … A G n 4. 81, 27, 9, 3, 1, … A G n
5. 2, 5, 10, 17, 26, . . . A G n 6. 1, [pic], 2, [pic] , 3, . . . A G n
When the terms of a sequence are added, the result is a _________________. A series can either be finite or infinite.
Finite series 2 + 4 + 6 + 8 Infinite series 2 + 4 + 6 + 8 + …
Series – we can use [pic] which is called ________________ or ______________.
[pic]
Trick: y= (the expression) Go to the table. Grab the values between 1 and 5. Add them together.
General Form for the Sum of the First n Terms in an Arithmetic Series: [pic],
where…
a1 is the ________ term
an is the ________term
n is the ___________ of terms
Ex. Find the sum of the arithmetic series[pic].
A 14
B 420
C 870
D 1568
General Form for the Sum of the First n Terms of a Geometric Series:
[pic], r ( 1
where
a1 is the _______ term
r is the ______________
n is the __________ of terms
Ex . Consider the following geometric series. Find the sum of the first ten terms.
|a) What is the sum of the first ten terms of 4 + 2 + 1 + ½ + …? |b) What is the value of the series below? [pic] |
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| |A [pic] |
| |B [pic] |
| |C [pic] |
| |D [pic] |
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7. Find [pic]. A or G ? [pic] or [pic]
8. Find [pic]. A or G ? [pic] or [pic]
Choose the best answer.
9. What is the third term of the sequence defined by an = 5n + 7? ________
10. What is the next term in the sequence? 0, 2, 6, 12, 20 . . . ? ________
11. Write out the series [pic]? __________________________________
[pic]
[pic]
12.4 Notes - Infinite Geometric Series
Infinite Geometric Series: the sum of an infinite series that has a _________
An infinite geometric series may have a ___________, depending on the value of the common ratio.
The Sum of an Infinite Geometric Series: [pic],
where
a1 is the _________ term of the series
r is the ____________________ and ____________
( If |r| > 1, the series _______________________________
Ex 2 Does the infinite geometric series have a sum?
|a) [pic] |b) [pic] |c) [pic] |
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|[pic] [pic] |[pic] [pic] |[pic] [pic] |
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|a)_________________ |b)________________ |c)_________________ |
Ex 3 Find the sum of the infinite geometric series if it has one.
|a) [pic] |b) [pic] |c) [pic] |
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|[pic] [pic] |[pic] [pic] |[pic] [pic] |
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| | |A -2 |
| | |B -½ |
| | |C 2 |
|a)_________________ |b)________________ |D The series has no sum. |
12.1-12.3 Sequences and Series Homework Day 1
[pic]
Is the sequence Arithmetic, Geometric or neither?
If it is A or G then find [pic]by using either [pic] or [pic].
8. -2, 0, 2, 4, … A G n 9. 3, 6, 12, 24, … A G n
10. 40, 10, [pic], [pic], … A G n 11. 4, 7, 12, 19, … A G n
[pic] [pic] [pic]
[pic] [pic] [pic]
12.4 Sequences and Series HW Day 2
For #’s 1 – 3, find the sum of the infinite geometric series, if it exists.
[pic] [pic]
[pic]
4. Tell whether the sequence -7, -1, 5, 11, 17, 23, … is arithmetic.
5. Find the sum of the arithmetic series [pic].
6. Tell whether the sequence [pic], 4, 24, 144, 864, … is geometric.
Algebra 2
Sequences & Series Review
Ch 12.1-12.4
TAKE OUT ALL YOUR NOTES FROM CHAPTER 12
Write next the formula what it is used to find.
[pic] [pic]
[pic] [pic] [pic]
Ques. 1 – 2. Is each sequence arithmetic, geometric or neither?
|108, 54, 27, 13.5… |2. 19, 13, 7, 1, -5,… |
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Ques. 3 – 4. Find the common difference or the common ratio.
| -1, 4, -16, 64, … |4. 24, 29, 34, 39 … |
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Question 5. Find [pic].
| 5, 13, 21, 29, … |
|(Hint: Is it arithmetic or geometric?) Formula_________________ |
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Question 6. Find [pic].
| 3, 12, 48, 192, … |
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|(Hint: Is it arithmetic or geometric?) Formula_________________ |
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Questions 7 – 8. Evaluate each sum.
|[pic] |[pic] |
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Question 9. Find the sum of the first ten terms.
| 11, 20, 29, … |
|[pic](2 + 9i ) |
|[pic] |
|[pic] |
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Questions 10 – 11. Find each sum if it exists. Leave answers as reduced fractions.
|10. [pic] |11. [pic] |
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|[pic] r = _______ |[pic] r = _______ |
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