Ch. 4 Flashcards Preview

Math II > Ch. 4 > Flashcards

Flashcards in Ch. 4 Deck (43)
Loading flashcards...
1

formulas to know

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

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

explicit geometric → an = a1rn-1

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

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

sum finite geometric series → S = a- anr / 1-r

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

2

sequence

an ordered list of numbers

4

terms

the numbers in a sequence

6

finite sequence

a sequence with a last term

8

infinite sequence

a sequence with no last term

10

graphing sequences

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

12

limit of a sequence

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

14

explicit formula

gives the value of any term an in terms of n

16

finding explicit formulas

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. || 

an = 90 - (n - 1)7

18

subscript 0

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.

20

percentage explicit formulas

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

        d1 = 10,000(1.1)

        d2 = d1(1.1)

             = 10,000(1.1)(1.1)

             = 10,000(1.1)2

        d3 = d2(1.1)

             = 10,000(1.1)2(1.1)

             = 10,000(1.1)3

                      •

                      •

                      •

         dn = 10,000(1.1)n

basically, x(1 + rate)n

         

22

self-similarity

the appearance of any part is similar to the whole thing

24

recursive formula

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. an = 2an-1   ⇔ recursion equation

26

recursion equation

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

28

finding recursive formulas

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

an = nan-1

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

a1 = 1

an = nan-1

30

percentage recursive formulas

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

an = (0.26)(an-1) + 650

  1. Use a calculator

Enter a1 → 650

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

Keep pressing Enter

sequence appears to approach limit of about 878

32

arithmetic sequence

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

34

common difference

d

the constant value an - an-1

36

geometric sequence

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

38

common ratio

r

the constant value an/an-1

40

explicit arithmetic formula

an = a1 + (n - 1)d

42

recursive arithmetic formula

a1 = value of 1st term

an = an-1 + d

44

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

Ex: 33, 29, 25, 21,..., 1

  1. arithmetic, geometric, or neither?

d = -4    →     arithmetic

  1. Use a formula

an = a1 + (n - 1)d

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

1 = 33 - 4n + 4

-36 = -4n

9 = n

there are 9 terms

46

explicit geometric formula

an = a1rn-1

48

recursive geometric formula

a1 = value of 1st term

an = (an-1)d

50

geometric mean

sqrt(ab)

Ex: find x in geo sequence 3, x, 18

  1. in a geo sequence, an/an-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)

52

series

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

 

54

finite series

has a last term

56

infinite series

has no last term

58

arithmetic series

its terms form an arithmetic sequence