Lesson 10.2: Arithmetic Sequences & Series Algebraic formula specific terms Recursion or Recursive formulas Arithmetic means Series Sum of finite arithmetic sequences

Arithmetic Sequence: a sequence whose consecutive terms have a common difference, d. d = 4 1. 5, 9, 13, 17, _____ d = 2. 2, -5, -12, -19, _____ The formula for an can be written given a1 and d: an = a1 + (n – 1)d = a1 + dn – d = dn + (a1 – d) simplified form an = dn + c (where c = a1 – d) For 1: a1 = 5, d = 4, so an = 5 + (n – 1)4 = For 2: a1 = 2, d = -7, so an =

Write the explicit formula for an an = a1 + (n – 1)d an = dn + c (where c = a1 – d) Write the explicit formula for an 3. a3 = 63, d = 4 a3 = dn + c so, an = 4n + 51 63 = 4*3 + c 51 = c p675 #30. a5 = 190, a10 = 115 a10 = a5 + (10 – 5)d a5 = dn + c 190 = -15*5 + c 265 = c 115 = 190 + 5d -75 = 5d -15 = d find d so, an = -15n + 265

Write the first 5 terms of the arithmetic sequence. (need the formula or a1 & d) an = a1 + (n – 1)d an = dn + c (where c = a1 – d) 1. an = 4n + 51 Have formula, so use it! a1= 4(1) + 51 = 55 a2 = 4(2) + 51 = 59 etc. Sequence: p675 #39. a8 = 26, a12 = 42 No formula, so find a1 & d a1 = -2 a2 = a1 + d = -2 + 4 = 2 a3 = a2 + d = a4 = a3 + d = a5 = a4 + d = a12 = a8 + (12 – 8)d a8 = a1 + (n – 1)d find d find a1 Sequence:

A recursion formula defines an in terms of previous term(s). For arithmetic sequences, an = an-1 + d (must also provide a term in the sequence). d = 4 1. 5, 9, 13, 17, _____ 21 a1 = 5, an = an-1 + 4 2. 2, -5, -12, -19, _____ - 26 d = -7 3. a3 = 63, d = 4 p674 #30. a5 = 190, a10 = 115 d = -15

1. Find the 4 arithmetic means between 72 and 52. Arithmetic means are terms between any nonconsecutive terms in an arithmetic sequence. an = a1 + (n – 1)d an = dn + c (where c = a1 – d) Need d. 1. Find the 4 arithmetic means between 72 and 52. 72, , , , , 52 a1 a6 = a1 + (n – 1)d a6 52 = 72 + (5)d -20 = 5d -4 = d So, 72, 68, 64, 60, 56, 52 2. Find the 3 arithmetic means between 21 and 45. d = 6, so 21, 27, 33, 39, 45

Arithmetic series: the sum of a finite arithmetic sequence. The sum of a finite arithmetic sequence is or See derivation on p670. note: an = a1 + (n – 1)d Examples: 1. Find the sum of the 1st 20 even integers beginning at 8. Have n = , a1 = , d = , so use 2nd formula. 2. Find Formula looks like an arithmetic series, expand a few terms to be sure: 2*3 + 1 = 7, 2*4 + 1 = 9, 2*5 + 1 = 11, … be careful! Can easily determine n, a1, & an, so use 1st formula. n = a1 = [k = 3] a108 = [k = 110]

Homework: Arithmetic Sequences: pp 674-675 # 2, 5, 7, 20, 24, 34, 38, 62, 64 Arithmetic Series: p675 #42, 44, 48, 50, 54, 66, 70, 72