Series NOTES Name ____________________________ Arithmetic Sequences

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.

USING AND WRITING SEQUENCES DOMAIN: 1 2 3 4 5 The domain gives the relative position of each term. The range gives the terms of the sequence. RANGE: 3 6 9 12 15 This is a finite sequence having the rule an = 3n, where an represents the nth term of the sequence.

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

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

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

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

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 2. (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

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

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

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:

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

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

Given an arithmetic sequence with X = 80

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 = -1 + 36 a10 = 35

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

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 = -2 + 5n OR an = 5n – 2

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…

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

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

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

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

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

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

Geometric Sequences

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

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

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:

-3, ____, ____, ____

r = a1= n = 9

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

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-1 an = 4 * 3n-1

Find the explicit formula for the following geometric sequence: 3, 6, 12, 24… an = a1*rn-1 a1 = 3 r =2 an = 3 *2n-1

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

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

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

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

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

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

Series

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

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 3 + 6 + 9 + 12 + 15 INFINITE SEQUENCE INFINITE SERIES 3, 6, 9, 12, 15, . . . 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 3 + 6 + 9 + 12 + 15 = ∑ 3i 5 i = 1

# of Terms: B – A + 1 UPPER BOUND TERM NUMBER SIGMA NTH TERM (SUM OF TERMS) NTH TERM SEQUENCE (EXPLICIT FORMULA) LOWER BOUND TERM NUMBER

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

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, …

Find the sum of the 1st Examples: 100 natural numbers. 1 + 2 + 3 + 4 + … + 100

S14 = a14 = 2 + (14 - 1)(3) = 41 Find the sum of the 1st Examples: 14 terms of the series: 2 + 5 + 8 + 11 + 14 + 17 +… To find a14 , you need a14 = 2 + (14 - 1)(3) = 41 S14 =

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

Finding the Sum from Summation Notation n = 4 a1 = 3 a4 = 6 3, 4, 5, 6 n = (7 – 4) + 1 a4 = 8 a7 = 14 8, 10, 12, 14

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

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:

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”

Sums of Infinite Series Made Finite (referred to as partial sums) Finding the Sum of Infinite Sequences “Converges” vs. “Diverges”

Find the sum, if possible: Geometric ~need to find r~ Is -1 < r < 1? Yes (Infinite Series - converges)

Find the sum, if possible: Is -1 < r < 1? No (Infinite series - Diverges)

Find the sum, if possible: Is -1 < r < 1? Yes (Infinite Series – Converges)

Find the sum, if possible: Is -1 < r < 1? No (Infinite Series–Diverges)

Find the sum, if possible: -1 < r < 1 Yes (Infinite Series–Converges)

Finding the Sum from Sigma Notation so “converges”

4 n=1 1st term 4th term Arithmetic, d= 3 Rewrite using sigma notation: 3 + 6 + 9 + 12 1st term n=1 4th term 4 Arithmetic, d= 3 Explicit formula

Geometric, r = ½ 5 n=1 1st term 5th term Explicit formula Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1 n=1 1st term 5 5th term Geometric, r = ½ Explicit formula

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