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Series NOTES Name ____________________________ Arithmetic Sequences

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**USING AND WRITING SEQUENCES**

The numbers in sequences are called terms. You can think of a sequence as a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1.

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**USING AND WRITING SEQUENCES**

DOMAIN: The domain gives the relative position of each term. The range gives the terms of the sequence. RANGE: This is a finite sequence having the rule an = 3n, where an represents the nth term of the sequence.

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**Write the first six terms of the sequence an = 2n + 3.**

Writing Terms of Sequences Write the first six terms of the sequence an = 2n + 3. SOLUTION a 1 = 2(1) + 3 = 5 1st term a 2 = 2(2) + 3 = 7 2nd term a 3 = 2(3) + 3 = 9 3rd term a 4 = 2(4) + 3 = 11 4th term a 5 = 2(5) + 3 = 13 5th term a 6 = 2(6) + 3 = 15 6th term

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**f (1) = (–2) 1 – 1 = 1 f (2) = (–2) 2 – 1 = –2 f (3) = (–2) 3 – 1 = 4**

Writing Terms of Sequences Write the first six terms of the sequence f (n) = (–2) n – 1 . SOLUTION f (1) = (–2) 1 – 1 = 1 1st term f (2) = (–2) 2 – 1 = –2 2nd term f (3) = (–2) 3 – 1 = 4 3rd term f (4) = (–2) 4 – 1 = – 8 4th term f (5) = (–2) 5 – 1 = 16 5th term f (6) = (–2) 6 – 1 = – 32 6th term

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**d = 3 r =2 r = d = -8 r = d = .4 d = r = ARITHMETIC ADD**

An introduction………… d = 3 r =2 r = d = -8 r = d = .4 d = r = ARITHMETIC ADD (by the same #) To get the next term GEOMETRIC MULTIPLY (by the same #) To get the next term

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**Finite VS. Infinite an-1 previous term an+1 next term**

Vocabulary of Sequences (Universal) an-1 previous term an+1 next term Finite VS. Infinite

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**The terms have a common difference of 2. (known as d)**

Arithmetic Sequence: sequence whose consecutive terms have a common difference. Example: 3, 5, 7, 9, 11, 13, ... The terms have a common difference of (known as d) To find the common difference you use an+1 – an Example: Is the sequence arithmetic? If so, find d. –45, –30, –15, 0, 15, 30 d = 15

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**Next four terms…… 12, 19, 26, 33 Find the next 4 terms of –9, -2, 5, …**

7 is referred to as d Next four terms…… 12, 19, 26, 33

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**Find the next four terms of 0, 7, 14, …**

Find the next four terms of x, 2x, 3x, … Find the next four terms of 5k, -k, -7k, … Arithmetic Sequence, d = 7 21, 28, 35, 42 Arithmetic Sequence, d = x 4x, 5x, 6x, 7x Arithmetic Sequence, d = -6k -13k, -19k, -25k, -31k

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4, 10, 16, 22 The nth term of an arithmetic sequence is given by: The nth term in the sequence The common difference The term # First term Find the 10th term:

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Find the 14th term of the sequence: Examples: 4, 7, 10, 13,……

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Examples: In the arithmetic sequence 4,7,10,13,…, which term has a value of 301?

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**Given an arithmetic sequence with**

X = 80

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Example: If the common difference is 4 and the fifth term is 15, what is the 10th term of an arithmetic sequence? an = a1 + (n – 1)d d = 4, a5 = 15, n = 5, a1=? 15 = a1 + (5 – 1)4 15 = a1 +16 a1 = –1 a10 = –1 + (10 – 1)4 = a10 = 35

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**Ex: 4, 6, 8, 10… Explicit vs. Recursive Formulas**

Explicit Formula – used to find the nth term of the arithmetic sequence in which the common difference and 1st term are known. Ex: 4, 6, 8, 10… Use a1 and d in sequence formula: an = 4 + (n – 1)2 an = 2n + 2

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**Find the explicit formula for the following arithmetic sequence:**

3, 8, 13, 18… an = a1 + (n – 1)d a1 = 3 d = 5 n = ? an = 3 + (n – 1)5 an = 3 + 5n – 5 an = n OR an = 5n – 2

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**an = an-1 + 2 an = an-1 + d a1 = ___ an+1 = an + d a1 = 4**

Explicit vs. Recursive Formulas Recursive Formula – (includes a1) used to find the next term of the sequence by adding the common difference to the previous term. an = an-1 + d a1 = ___ an+1 = an + d an = an-1 + 2 a1 = 4 Ex: 4, 6, 8, 10…

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**a1 = 3 an = an-1 + d a1 = 3 d = 5 an = an-1 + 5**

Series NOTES Name ____________________________ Find the recursive formula for the following arithmetic sequence: 3, 8, 13, 18… an = an-1 + d a1 = 3 d = 5 an = an-1 + 5 a1 = 3

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**an = an-1 + 6 a1 = 4 Using Recursive & Explicit Formulas**

1. Create the 1st 5 terms: 4, 10, 16, 22, 28 a2 = = 10 2. Find the explicit formula: an = a1 + (n – 1)d a3 = = 16 an = 4 + (n – 1)6 a4 = = 22 an = 4 + 6n – 6 an = 6n – 2 a5 = = 28

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**Using Recursive & Explicit Formulas**

an = 7 – 2n 1. Create the 1st 5 terms: 5, 3, 1, –1, –3 a1 = 7 – 2(1) = 5 a2 = 7 – 2(2) = 3 2. Find the recursive formula: a3 = 7 – 2(3) = 1 a4 = 7 – 2(4) = –1 an = an-1 – 2 a1 = 5 a5 = 7 – 2(5) = –3

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**Examples: Insert 3 arithmetic means between 8 & 16.**

An arithmetic mean of two numbers, a and b, is simply their average. Use the formula and information given to find the common difference to create the sequence. Examples: Insert 3 arithmetic means between 8 & 16. 14 10 12 Let 8 be the 1st term Let 16 be the 5th term Let 5 be N d is missing

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**The two arithmetic means are –1 and 2,**

Find two arithmetic means between –4 and 5 -4, ____, ____, 5 The two arithmetic means are –1 and 2, since –4, -1, 2, 5 forms an arithmetic sequence

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**Find 3 arithmetic means between 1 & 4**

1, ____, ____, ____, 4 The 3 arithmetic means are since 1, ,4 forms a sequence

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Geometric Sequences

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**Finite VS. Infinite an-1 previous term an+1 next term**

Series NOTES Name ____________________________ Vocabulary of Sequences (Universal) an-1 previous term an+1 next term Finite VS. Infinite

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**Use to determine common ratio**

Find the next 3 terms of 2, 3, 9/2, __, __, __ 3 – 2 vs. 9/2 – 3… not arithmetic Use to determine common ratio

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**1st term: 2 4th term: 5th term: 6th term: How is the formula derived?**

The nth term of a geometric sequence is given by: 1st term: 2 4th term: 5th term: 6th term:

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-3, ____, ____, ____

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r = a1= n = 9

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)2-1 ( 2 8 - = a x =

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**Ex: 4, 12, 36, 108… Use a1 and r in sequence formula:**

Explicit vs. Recursive Formulas Explicit Formula – used to find the nth term of the geometric sequence in which the common ratio and 1st term are known. Ex: 4, 12, 36, 108… Use a1 and r in sequence formula: Ex: an = a1*rn an = 4 * 3n-1

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**Find the explicit formula for the following geometric sequence:**

3, 6, 12, 24… an = a1*rn a1 = 3 r =2 an = 3 *2n-1

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**an+1 = r(an) an = an-1 (r) a1 = ___ an = an-1 (–4) a1 = –1 a1 (r) = a2**

Explicit vs. Recursive Formulas Recursive Formula (includes a1) – used to find the next term of the sequence by multiplying the common ratio to the previous term. an = an-1 (r) a1 = ___ an+1 = r(an) Ex: –1, 4, –16, 64 … a1 (r) = a2 an = an-1 (–4) a1 = –1 a2 (r) = a3 a3 (r) = a4

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**a1 = 3 an = an-1 * r a1 = 3 r = 2 an = an-1 * 2**

Series NOTES Name ____________________________ Find the recursive formula for the following geometric sequence: 3, 6, 12, 24… an = an-1 * r a1 = 3 r = 2 an = an-1 * 2 a1 = 3

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**an = an-1 (3) a1 = –1 Using Recursive & Explicit Formulas**

1. Create the 1st 5 terms: –1, –3, –9, –27, – 81 a2 = –1(3) = –3 2. Find the explicit formula: an = a1 (r)n-1 a3 = –3(3) = –9 an = –1(3)n-1 a4 = –9(3) = –27 an = –3n-1 a5 = –27(3) = –81

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**Using Recursive & Explicit Formulas**

an = 2(4)n – 1 1. Create the 1st 5 terms: 2, 8, 32, 128, 512 a1 = 2(4)1-1 = 2 a2 = 2(4)2-1 = 8 2. Find the recursive formula: a3 = 2(4)3-1 = 32 a4 = 2(4)4-1 = 128 an = 4an-1 a1 = 2 a5 = 2(4)5-1 = 512

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**6 –18 Ex: Find two geometric means between –2 and 54**

A geometric mean(s) of numbers are the terms between any 2 nonsuccessive terms of a geometric sequence. Use the terms given to find the common ratio and find the missing terms called the geometric means. Ex: Find two geometric means between –2 and 54 The 2 geometric means are 6 and -18 6 –18 -2, ____, ____, 54

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***** Insert one geometric mean between ¼ and 4*****

Series NOTES Name ____________________________ *** Insert one geometric mean between ¼ and 4*** *** denotes trick question

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Series

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**Finite VS. Infinite an-1 previous term an+1 next term**

Series NOTES Name ____________________________ Vocabulary of Sequences (Universal) an-1 previous term an+1 next term Finite VS. Infinite

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USING SERIES When the terms of a sequence are added, the resulting expression is a series. A series can be finite or infinite. FINITE SEQUENCE FINITE SERIES 3, 6, 9, 12, 15 INFINITE SEQUENCE INFINITE SERIES 3, 6, 9, 12, 15, . . . . . . You can use summation notation to write a series. For example, for the finite series shown above, you can write = ∑ 3i 5 i = 1

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**# of Terms: B – A + 1 UPPER BOUND TERM NUMBER SIGMA NTH TERM**

(SUM OF TERMS) NTH TERM SEQUENCE (EXPLICIT FORMULA) LOWER BOUND TERM NUMBER

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**It can be infinite or finite.**

An arithmetic series is a series associated with an arithmetic sequence. It can be infinite or finite. Definition:

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**1, 4, 7, 10, 13, …. No Sum 3, 7, 11, …, 51 Infinite Arithmetic**

(constantly getting larger or smaller) 1, 4, 7, 10, 13, …. No Sum 3, 7, 11, …, 51 Finite Arithmetic 1, 2, 4, …, 64 1, 2, 4, 8, …

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**Find the sum of the 1st Examples: 100 natural numbers.**

… + 100

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**S14 = a14 = 2 + (14 - 1)(3) = 41 Find the sum of the 1st Examples:**

14 terms of the series: … To find a14 , you need a14 = 2 + (14 - 1)(3) = 41 S14 =

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Examples: Find the sum of the series Need 13th term: 4(13) + 5 = 57

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**Finding the Sum from Summation Notation**

n = a1 = a4 = 6 3, 4, 5, 6 n = (7 – 4) a4 = a7 = 14 8, 10, 12, 14

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19, 23, 27, 31…79 a4 = a19 = n = (19 - 4) + 1 = 16 15, 17, 19, …47 a7 =15 a23 = n = (23-7) + 1 = 17

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**An geometric series is a series associated with a geometric sequence.**

They can be infinite or finite. Finite and infinite have different formulas depending on the value of r. Definition:

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**No Sum No Sum 1, 2, 4, …, 64 1, 2, 4, 8, … Finite Geometric**

1, 4, 7, 10, 13, …. Infinite Arithmetic (constantly getting larger or smaller) No Sum 3, 7, 11, …, 51 Finite Arithmetic 1, 2, 4, …, 64 Finite Geometric Infinite Geometric r < -1 OR r > 1 (constantly getting larger or smaller) “diverges” 1, 2, 4, 8, … No Sum Infinite Geometric -1 < r < 1 “converges”

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**Sums of Infinite Series Made Finite (referred to as partial sums) **

Finding the Sum of Infinite Sequences “Converges” vs. “Diverges”

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**Find the sum, if possible:**

Geometric ~need to find r~ Is -1 < r < 1? Yes (Infinite Series - converges)

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**Find the sum, if possible:**

Is -1 < r < 1? No (Infinite series - Diverges)

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**Find the sum, if possible:**

Is -1 < r < 1? Yes (Infinite Series – Converges)

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**Find the sum, if possible:**

Is -1 < r < 1? No (Infinite Series–Diverges)

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**Find the sum, if possible:**

-1 < r < Yes (Infinite Series–Converges)

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**Finding the Sum from Sigma Notation**

so “converges”

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**4 n=1 1st term 4th term Arithmetic, d= 3**

Rewrite using sigma notation: 1st term n=1 4th term 4 Arithmetic, d= 3 Explicit formula

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**Geometric, r = ½ 5 n=1 1st term 5th term Explicit formula**

Rewrite using sigma notation: n=1 1st term 5 5th term Geometric, r = ½ Explicit formula

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**SIGMA NOTATION: NUMERATOR: DENOMINATOR:**

Rewrite the following using sigma notation: Numerator is geometric, r = 3 Denominator is arithmetic d= 5 NUMERATOR: DENOMINATOR: SIGMA NOTATION:

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