Got ID? 1-9-17 & 1-10-17 AM7.1a To Identify an Arithmetic or Geometric Sequence and to Define Sequences and Series (Get the note taking guide from the front. Pay careful attention to these definitions.)

LESSON : Sequence: a set of numbers arranged in a specific order. The terms of a sequence are the numbers in it. The notation: t1, t2, t3, … is used to designate a sequence. If it has three dots . . . that means it is an infinite sequence. If it just ends, it is a finite sequence. The subscript refers to the position of the term. The term tn refers to the nth place or general term.

* (difference = 4) (difference = -7) (difference = d) Arithmetic Sequence: a listing of numbers whereby the difference of any two consecutive terms is constant. This is called the common difference. Example: 2, 6, 10, 14, 18 . . . 17, 10, 3, -4, -11 . . . a, a + d, a + 2d, a + 3d, a + 4d . . . (difference = 4) (difference = -7) (difference = d) To find the difference, subtract the second term by the first, or any term by the previous one. Ex: 10 – 17 = -7 (d = 2nd – 1st, 3rd – 2nd, etc.) *

Geometric Sequence: a listing of numbers whereby the ratio of any two consecutive terms is constant. This is called the common ratio. Example: 1, 3, 9, 27, 81 . . . 64, -32, 16, -8, 4 . . . a, ar, ar2, ar3, ar4 . . . (ratio = 3) (ratio = -½) (ratio = r) To find the ratio, divide the second term by the first, or any term by the previous one. Ex: -32/64 = -½ *

Class activity: (From page 477 in your book.) For problems 17 to 27 odds, determine whether the sequence is Arithmetic, Geometric, or neither. If it is Arithmetic, find the difference. If it is Geometric, find the ratio. 17, 21, 25, 29, . . . 8, 12, 18, 27, . . . 21. 1, 4, 9, 16, . . . 23. 11, 101, 1001, 10001, . . . 25. 2a – 2b, 3a – b, 4a, 5a + b, Arithmetic, d = 4 Geometric, r = 3/2 Neither Neither Arithmetic, d = a + b Geometric, r = 2

* * tn = t1 + (n – 1)*d t1 = the first term. tn = the nth term. Let’s look at the first Arithmetic example again: 2, 6, 10, 14, 18 . . . could be represented by: 2 + 0*4, 2 + 1*4, 2 + 2*4, 2 + 3*4, 2 + 4*4, … 1st term 2nd term 3rd term 4th term 5th term And the last one: a, a + d, a + 2d, a + 3d, a + 4d . . . could be represented by a +0*d, a + 1*d, a + 2*d, a + 3*d, a + 4*d, . . . See a pattern? 1st term 2nd term 3rd term 4th term 5th term Can we come up with a formula for an Arithmetic Sequence? * tn = t1 + (n – 1)*d * t1 = the first term. tn = the nth term. d = difference. n = the position in the sequence.

tn = t1 + (n – 1)d tn = t19 t1 = 11 t19 = 11 + (19 – 1)(-2) Ex: Find the 19th term in the arithmetic sequence for which t1 = 11 and t2 is 9. tn = t1 + (n – 1)d (First, identify the variables.) (Fill in the variables.) tn = t19 t1 = 11 n = 19 d = 9 – 11 = -2 t19 = 11 + (19 – 1)(-2) t19 = 11 + (18)(-2) t19 = 11 + (-36) t19 = -25

tn = t1 + (n – 1)d tn = t68 t1 = 3 t68 = 3 + (68 – 1)(5) Try: Find the 68th term in the arithmetic sequence for which t1 = 3 and t2 is 8. tn = t1 + (n – 1)d (First, identify the variables.) (Fill in the variables.) tn = t68 t1 = 3 n = 68 d = 8 – 3 = 5 t68 = 3 + (68 – 1)(5) t68 = 3 + (67)(5) t68 = 3 + (335) t68 = 338

Let’s look at the first Geometric example again: 1, 3, 9, 27, 81, . . . = 1*30, 1*31 , 1*32, 1*33, 1*34, . . . 1st 2nd 3rd 4th 5th See a pattern? And the last one: a, ar, ar2, ar3, ar4, . . . = ar0, ar1, ar2, ar3, ar4, . . . 1st 2nd 3rd 4th 5th Can we come up with a formula for a Geometric Sequence? * tn = t1 * r(n – 1) * t1 = the first term. tn = the nth term. r = ratio (keep r as a ratio (fraction)!!!). n = the position in the sequence.

* tn = t1 * r(n – 1) (r is a ratio) tn = t5 t1 = 8 Try: Find the 5th term in the geometric sequence for which t1 = 8 and t2 is 12. tn = t1 * r(n – 1) (r is a ratio) (First, identify the variables.) (Fill in the variables.) tn = t5 t1 = 8 r = 12/8 = 3/2 n = 5 * (*Leave as a ratio)

* tn = t1 * r(n – 1) (r is a ratio) tn = t100 t1 = Try: Find the 100th term in the geometric sequence for which t1 = and t2 is . tn = t1 * r(n – 1) (r is a ratio) (First, identify the variables.) (Fill in the variables.) tn = t100 t1 = r = n = 100 * (*Leave as a ratio)

Active Learning Assignment: Arithmetic (#15, 16, 19, 20 ,21, 23, 25, 26) & Geometric Handout (#12, 13, 16, 18, 19) Quiz on these tomorrow. WOW: Become the most positive and enthusiastic person you know!