Sequences A2/trig

Sequences: Vocabulary Sequence: an ordered list of numbers Ex. -2, -1, 0, 1, 2, 3 Term: each number in a sequence Ex. a1, a2, a3, a4, a5, a6 Recursive Formula: finding the next term depends on knowing a term or terms before it. Explicit formula: defines the nth term of a sequence.

Vocabulary Recursive Formula: an-1 Uses one or more previous terms to generate the next term. Any term: an Previous term: an-1 an-1

Examples: A) Write the first six terms of the sequence where a1 = -2 and an = 2an-1 – 1 ( Always list the terms with subscripts first:) a1, a2, a3, a4, a5, a6 B) Write the first six terms of the sequence where a1 = 4 and an = 3an-1 + 5

Examples: A) Write the first six terms of the sequence where a1 = -2 and an = 2an-1 – 1 a1, a2, a3, a4, a5, a6 -2,-5,-11,-23,-47,-95 B) Write the first six terms of the sequence where a1 = 4 and an = 3an-1 + 5 a1, a2, a3, a4, a5, a6 4, 17, 56 ,173, 524, 1577

Recursive formula depending on two previous terms: a1=-2, a2 = 3 ak= ak-1 + ak-2 Find the first 6 terms.

Recursive formula depending on two previous terms: a1=-2, a2 = 3 ak= ak-1 + ak-2 Find the first 6 terms. a1, a2, a3, a4, a5, a6 -2, 3, 1, 4, 5, 9

Some sequences are neither of these! Two special sequence types: Arithmetic sequence:a sequence in which each term is found by adding a constant, called the common difference (d), to the previous term. Geometric sequence: a sequence in which each term is founds by multiplying a constant, (r), called a common ratio to the previous term. Some sequences are neither of these!

Do now: Recursive formula Find the 10th term of a1 = 7 and an = an-1 + 6 Recursive formula

Example 1: Find the 10th term of a1 = 7 and an = an-1 + 6 7,13,19,25,31,37,43,49,55,61

Formula for the nth term Common difference an = a1 + (n – 1)d First term in the sequence What term you are looking for What term you are looking for

Example: Find the 10th term of a1 = 7 and an = an-1 + 6 Write the explicit formula (recall a10 = 61) an = a1 + d(n – 1) an= 7+ 6(n-1) an = 7 + 6n – 6 an = 6n + 1 a10 = 6(10) +1 = 61

Vocabulary Arithmetic Sequence: Start by asking, What is d? a2 – a1 A sequence generated by adding “d” a constant number to pervious term to obtain the next term. This number is called the common difference. Start by asking, What is d? a2 – a1 3, 7, 11, 15, … d = 4 8, 2, -4, -10, … d = -6

Find the explicit formula for these examples: For Arithmetic Sequences, use the formula: an = a1 + d(n – 1) 3, 7, 11, 15, … d = 4 8, 2, -4, -10, … d = -6

solutions: an = a1 + d(n – 1) an = 4n – 1 an = -6n + 14

Examples when a1 is not given Find the 10th term of the arithmetic sequence where a3 = -5 and a6 = 16 Find the 15th term of the arithmetic sequence where a5 = 7 and a10 = 22 Find the 12th term of the arithmetic sequence where a3 = 8 and a7 = 20

Examples when a1 is not given Find the 10th term of the arithmetic sequence where a3 = -5 and a6 = 16 16- -5 =21 6-3 = 3 an = -19 + 7(n – 1) an = -19 + 7n – 7 an = 7n – 26 A10 = 7(10)-26=44

Examples when a1 is not given Find the 15th term of the arithmetic sequence where a5 = 7 and a10 = 22 22-7 = 15 10-5 = 5 a1=-5 a2=-2 a3=1 a4=4 a5=7 an = -5 + 3(n – 1) an = -5 + 3n – 3 an = 3n – 8 A15 = 3(15)-8=37

Examples when a1 is not given Find the 12th term of the arithmetic sequence where a3 = 8 and a7 = 20 D = 3 a1 = 2 an = 2 + 3(n – 1) an = 2 + 3n – 3 an = 3n – 1 A12 = 3(12)-1=35

Vocabulary Arithmetic Means: Terms in between 2 nonconsecutive terms Ex. 5, 11, 17, 23, 29  11, 17, 23 are the arithmetic means between 5 & 29

Example 3: Find the 4 arithmetic means between 10 & -30

Example 3: Find the 4 arithmetic means between 10 & -30 10, 2, -6, -14, -22 -30

Example 3: Find the 5 arithmetic means between 6 & 60 6, 15, 24, 33, 42, 51, 60 Example 3:

Geometric Sequences multiplying

Do Now: Find the 5th term of a1 = 8 and an = 3an-1

Do Now: Find the 5th term of a1 = 8 and an = 3an-1 8, 24, 72, 216, 648 5, 10, 20, 40, 80, 160, 320

Vocabulary Geometric Sequence: What is r? A sequence generated by multiplying a constant ratio to the previous term to obtain the next term. This number is called the common ratio. What is r? 2, 4, 8, 16, … r = 2 27, 9, 3, 1, … r = 1/3

Explicit Formula for the nth term First term in the sequence an = a1rn-1 What term you are looking for What term you are looking for Common Ratio

Explicit geometric formula an = a1rn-1 Find the Explicit formula and the 5th term of a1 = 8 and an = 3an-1 Find the Explicit formula and the 7th term of a1 = 5 and an = 2an-1

Explicit geometric formula Find the explicit formula and the 5th term of a1 = 8 and an = 3an-1 an = a1rn-1 an = 8(3)n-1 a5 = 8(3)4 = 648 Find the Explicit formula and the 7th term of a1 = 5 and an = 2an-1 an = a1rn-1 a7 = 5(2)6 = 320

Warm up 1. Find the 8th term of the sequence defined by a1= –4 and an= an-1+ 2 2. Find the 12th term of the arithmetic sequence in which a4= 2 and a7= 6 3. Find the four arithmetic means between 6 and 26. 4. Find the 5th term on the sequence defined by a1= 2 and an= 2an-1.

Summation

Series Series: the sum of a sequence Summation Notation: EX. (for the above series) End number Formula to use Start number

= 2(1)-1 + 2(2)-1 + 2(3) -1 + 2(4) -1 = 1 + 3 +5 + 7 =16

Summation Properties For sequences ak and bk and positive integer n:

Summation Formulas For all positive integers n: Constant Linear Quadratic

Example 1: Evaluate

Example 1: Evaluate

Extra Example: Evaluate

Extra Example: Evaluate

Sum of an arithmetic sequence Arithmetic Series Sum of an arithmetic sequence

Do Now: add the terms of the 4 series above Arithmetic Sequences Geometric Sequences ADD To get next term MULTIPLY To get next term Arithmetic Series Sum of Terms Geometric Series Sum of Terms Do Now: add the terms of the 4 series above

Do Now: add the terms of the 4 series above Arithmetic Series Sum of Terms Geometric Series Sum of Terms Arithmetic Sequences Geometric Sequences ADD To get next term MULTIPLY To get next term Do Now: add the terms of the 4 series above

Formula for arithmetic series Vocabulary An Arithmetic Series is the sum of an arithmetic sequence. Formula for arithmetic series Sn=

Example 1: Find the series 1, 3, 5, 7, 9, 11 B. Find the series 8, 13, 18, 23, 28, 33, 38

Example 1: Find the series 1, 3, 5, 7, 9, 11 B. Find the series 8, 13, 18, 23, 28, 33, 38

Example 2: Given 3 + 12 + 21 + 30 + …, find S25 Find the 25th and the 11th terms by finding the explicit formula first.

Example 2: Given 3 + 12 + 21 + 30 + …, find S25 Now apply series formula..

Example 2: Given 3 + 12 + 21 + 30 + …, find S25

Example 3: Evaluate

Formula for geometric series Vocabulary An Geometric Series is the sum of an geometric sequence. Formula for geometric series Sn=

Example 1: Given the series 3 + 4.5 + 6.75 + 10.125 + …find S5

Example 1: Given the series 3 + 4.5 + 6.75 + 10.125 + …find S5

Vocabulary of Sequences (Universal)

1/2 7 x 127/128

Do Now: Evaluate

Do Now: Evaluate

Do Now: n Evaluate r a1

Formula for a convergent infinite geometric series Vocabulary An Infinite Geometric Series is a geometric series with infinite terms. Formula for a convergent infinite geometric series S = If r <1 then the _______ can be found (converges) If r  1 then the _______ can’t be found (diverges) SUM SUM

Examples : Find the sum of the infinite geometric series 3 + 1.2 + 0.48 + 0.192 + … Find the partial sum (S4) Determine the ratio Find the sum of the infinite geometric series 8 + 9.6 + 11.52 + 13.824 + …

Example 1: Find the sum of the infinite geometric series 3 + 1.2 + 0.48 + 0.192 + … r = .4 B) Find the sum of the infinite geometric series 8 + 9.6 + 11.52 + 13.824 + … r= 1.2 so it is divergent

Example 2: Find the sum of the infinite geometric series below:

Example 2: Find the sum of the infinite geometric series below: r =

Example 3: Write 0.2 as a fraction in simplest form.

Example 3: Write 0.2 as a fraction in simplest form.