5-7-15 AM7.1a To Identify an Arithmetic or Geometric Sequence and to Define Sequences and Series (Be ready to copy as soon as the bell rings! Pay careful attention to these definitions—you will be tested on them.)

Ex: 10 – 17 = -7 (d = 2nd – 1st, 3rd – 2nd, etc.) OPENER & LESSON—Copy: 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: Open your books to page 477. 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

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.

* * 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, … 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, . . . 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

t1 = the first term. tn = the nth term. Let’s look at the first Geometric example again: 1, 3, 9, 27, 81, . . . = 1*30, 1*31 , 1*32, 1*33, 1*34, . . . And the last one: a, ar, ar2, ar3, ar4, . . . = ar0, ar1, ar2, ar3, ar4, . . . 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) 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: P 477: 29 – 38