Ch. 4

Question 1

formulas to know

Answer 1

explicit arithmetic → a_n = a₁ + d(n - 1)

recursive arithmetic → a₁ = first term; a_n = a_n-1 + d

explicit geometric → a_n = a₁r^n-1

recursive geometric → a₁ = first term; a_n = (a_n-1)r

sum finite arithmetic series → S = n(a₁₊a_n) / 2

sum finite geometric series → S = a₁- a_nr / 1-r

sum geometric infinite series → S = a₁ / 1-r; |r| < 1

Question 2

sequence

Answer 2

an ordered list of numbers

Question 4

terms

Answer 4

the numbers in a sequence

Question 6

finite sequence

Answer 6

a sequence with a last term

Question 8

infinite sequence

Answer 8

a sequence with no last term

Question 10

graphing sequences

Answer 10

each term is paired w/ a number that gives its position in the sequence

by plotting points with coord’s (position, term), u can graph a sequence on a coord plane

Question 12

limit of a sequence

Answer 12

when the terms of an infinite sequence get closer and closer to a single fixed number L, then L is called the limit

not all infinite sequences have limits; repeating ones, for example, like 1, 2, 3, 1, 2, 3,…

graphing it often helps

Question 14

explicit formula

Answer 14

gives the value of any term aⁿ in terms of n

Question 16

finding explicit formulas

Answer 16

Ex: Given 90, 83, 76, 69,…, find a formula for the sequence

1. Find a pattern. || a repeated subtraction of 7
2. Write the 1st few terms. Show how you found each term. ||

a₁ = 90

a₂ = 83 = 90 - 1(7)

a₃ = 76 = 90 - 2(7)

a₄ = 69 = 90 - 3(7)

1. Express the pattern in terms of n. ||

a_n = 90 - (n - 1)7

Question 18

subscript 0

Answer 18

when the 1st term of a sequence represents a starting value before any change ocurs, subscript 0 is often used

Ex: monthly ank account balances, 1st term is v₀ for initial deposit. Next term = v₁, for 1st month’s interest, etc. etc.

Question 20

percentage explicit formulas

Answer 20

Ex: bacteria count increases 10% each day; 10,000 now; Find formula for bacteria count after n days

1. d + .1d {if annual interest, compounded monthly, then d + (.1/12)d}

d(1 +.1)

d(1.1)

1. d₀ = 10,000

    d<sub>1</sub> = 10,000(1.1)
   
    d<sub>2</sub> = d<sub>1</sub>(1.1)
   
         = 10,000(1.1)(1.1)
   
         = 10,000(1.1)<sup>2</sup>
   
    d<sub>3</sub> = d<sub>2</sub>(1.1)
   
         = 10,000(1.1)<sup>2</sup>(1.1)
   
         = 10,000(1.1)<sup>3</sup>
   
                  •
   
                  •
   
                  •
   
     d<sub>n</sub> = 10,000(1.1)<sup>n</sup>

basically, x(1 + rate)ⁿ

Question 22

self-similarity

Answer 22

the appearance of any part is similar to the whole thing

Question 24

recursive formula

Answer 24

tells how to find the nth term from the term(s) before it

2 parts:

1. a₁ = 1 ⇔ value(s) of 1st term(s) given
2. a_n = 2a_n-1 ⇔ recursion equation

Question 26

recursion equation

Answer 26

shows how to find each term from the term(s) before it

Question 28

finding recursive formulas

Answer 28

Ex: 1, 2, 6, 24

1. Write several terms of the sequence using subscripts. Then look for the relationship b/w each term & the term before it

a₁ = 1

a₂ = 2 = 2 • 1

a₃ = 6 = 3 • 2

a₄ = 24 = 4 • 6

1. Write in terms of a

a₂ = 2 • a₁

a₃ = 3 • a₂

a₄ = 4 • a₃

1. Write a recursion equation

a_n = na_n-1

1. Use value of 1st term & recursion equation to write recursive formula

a₁ = 1

a_n = na_n-1

Question 30

percentage recursive formulas

Answer 30

Ex: 650mg of aspirin every 6h; only 26% of aspirin remaining in body by the time of new dose; what happens to amount of aspirin in body if taken several days?

1. Write a recursion equation

amount aspirin after nth dose = 26% amount after prev. dose + new dose of 650mg

a_n = (0.26)(a_n-1) + 650

1. Use a calculator

Enter a₁ → 650

Enter recursion equation using ANS for a_n-1 → .26ANS + 650

Keep pressing Enter

sequence appears to approach limit of about 878

Question 32

arithmetic sequence

Answer 32

a sequence in which the difference b/w any term & the term before it is a constant

Ex: 2, 4, 6, 8

+2, +2, +2

Question 34

common difference

Answer 34

d

the constant value a_n - a_n-1

Question 36

geometric sequence

Answer 36

a sequence in which the ratio of any term to the term before it is a constant

Ex: 2, 4, 8, 16

x2, x2, x2

Question 38

common ratio

Answer 38

r

the constant value a_n/a_n-1

Question 40

explicit arithmetic formula

Answer 40

a_n = a₁ + (n - 1)d

Question 42

recursive arithmetic formula

Answer 42

a₁ = value of 1st term

a_n = a_n-1 + d

Question 44

using explicit formulas to find out how many terms are in a finite sequence

Answer 44

Ex: 33, 29, 25, 21,…, 1

1. arithmetic, geometric, or neither?

d = -4 → arithmetic

1. Use a formula

a_n = a₁ + (n - 1)d

1 = 33 + (n - 1)(-4)

1 = 33 - 4n + 4

-36 = -4n

9 = n

there are 9 terms

Question 46

explicit geometric formula

Answer 46

a_n = a₁r^n-1

Question 48

recursive geometric formula

Answer 48

a₁ = value of 1st term a_n = (a_n-1)d

Question 50

geometric mean

Answer 50

sqrt(ab) Ex: find x in geo sequence 3, x, 18 1. in a geo sequence, a_n/a_n-1 is a constant a₂/a₁ = a₃/a₂ x/3 = 18/x x² = 54 x = +- sqrt(54) = +- 3sqrt(6) x is 3sqrot(6) or -3sqrt(6)

Question 52

series

Answer 52

the indicated sum of the terms when teh terms of a sequence are added sequence = 1,2,3,4,5,6 series = 1 + 2 + 3 + 4 + 5 + 6

Question 54

finite series

Answer 54

has a last term

Question 56

infinite series

Answer 56

has no last term

Question 58

arithmetic series

Answer 58

its terms form an arithmetic sequence

Question 60

sum finite arithmetic series

Answer 60

S = n(a₁ + a_n) \_\_\_\_\_\_\_\_\_ 2

Question 62

finding the sum of a finite series

Answer 62

Ex: 7 + 12 + 17 + 22 +...+ 52 1. arithmetic, geometric, or neither d = 5, arithmetic 1. Find the # of terms a_n = a₁ + (n -1)d 52 = 7 + (n - 1)5 52 = 7 + 5n - 5 50 = 5n 10 = n 10 terms 1. Use formula S = n(a₁ + a_n)/2 = 10(7 + 52)/2 = 295

Question 64

sigma notation

Answer 64

uses summation symbol Greek sigma Σ Read: The sum of 2n for integer values of n from 1 to 6 if infinite series, number on top is infinity symbol

Question 66

expanded form

Answer 66

when you substitute the values of n into the formula

Question 68

geometric series

Answer 68

its terms form a geometric sequence

Question 70

sum finite geometric series

Answer 70

S = a₁ - a_nr \_\_\_\_\_\_\_\_ r

Question 72

evaluating sigma notation

Answer 72

​if finite, 1. write in expanded form 2. decide if arithmetic, geometric, or neither 1. use formula for sum of whatever type it is if infinite, 1. see whether a sum even exists first by expanding first few terms 2. if geometric, use ratio to see if sum exists. if arithmetic, graph. 3. find sum using formula or graphing

Question 74

partial sum

Answer 74

the sum of the first n terms of an infinite series

Question 76

sequence of partial sums

Answer 76

2, (2 + 4), (2 + 4 + 6), ... forms a sequence of partial sums for the infinite series 2 + 4 + 6 +...

Question 78

sum of an infinite series

Answer 78

if the sequence of the partial sums of an infinite series has a limit, then that limit is the sum of the series

Question 80

finding sums of infinite series by graphing

Answer 80

Ex: 3 + (-1.5) + .75 + (-0.375) + ... 1. Find common ratio → r = 0.5 2. Find 1st few partial sums S₁ = 3 S₂ = 3 + (-1.5) = 1.5 S₃ = 1.5 + 0.75 = 2.25 S₄ = 2.25 + (-0.375) = 1.875 S₅ = 1.875 + .1875 = 2.0625 S₆ = 2.0625 + (-0.09375) = 1.96875 1. Graph the partial sums (position, term) 2. limit is about 2, so sum is about 2

Question 82

sum infinite geometric series

Answer 82

for any infinite geometric series with |r| \< 1, the terms get closer & closer to 0. This suggests substituting 0 for a_n in the formula for the sum of a finite geometric series S = a₁ - a_nr \_\_\_\_\_\_\_ 1 - r ► a₁ - 0 • r \_\_\_\_\_\_\_\_ 1 - r ► a₁/1 -r for |r| \< 1

Question 84

finding sums of infinite series by formula

Answer 84

Ex: 4 + 4/5 + 4/25 + 4/125 + ... 1. must be geometric with |r| \< 1 geometric; r = 1/5 1. Use formula for infintie geometric series S = a₁/1-r = 4/(1-1/5) = 5 1. the sum of the series is 5