Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

Math II > Ch. 4 > Flashcards

Ch. 4 Flashcards

1
Q

formulas to know

A

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 = a1- anr / 1-r

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

sequence

A

an ordered list of numbers

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

terms

A

the numbers in a sequence

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

finite sequence

A

a sequence with a last term

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

infinite sequence

A

a sequence with no last term

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

graphing sequences

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

limit of a sequence

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

explicit formula

A

gives the value of any term an in terms of n

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

finding explicit formulas

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

subscript 0

A

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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

percentage explicit formulas

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

self-similarity

A

the appearance of any part is similar to the whole thing

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

recursive formula

A

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
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
26
Q

recursion equation

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
28
Q

finding recursive formulas

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
30
Q

percentage recursive formulas

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
32
Q

arithmetic sequence

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
34
Q

common difference

A

d

the constant value an - an-1

36
Q

geometric sequence

A

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
Q

common ratio

A

r

the constant value an/an-1

40
Q

explicit arithmetic formula

A

an = a1 + (n - 1)d

42
Q

recursive arithmetic formula

A

a1 = value of 1st term

an = an-1 + d

44
Q

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

A

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
Q

explicit geometric formula

A

an = a1rn-1

48
Q

recursive geometric formula

A

a1 = value of 1st term

an = (an-1)d

50
Q

geometric mean

A

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
Q

series

A

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
Q

finite series

A

has a last term

56
Q

infinite series

A

has no last term

58
Q

arithmetic series

A

its terms form an arithmetic sequence

60
Q

sum finite arithmetic series

A

S =

n(a1 + an)

_________

2

62
Q

finding the sum of a finite series

A

Ex: 7 + 12 + 17 + 22 +…+ 52

  1. arithmetic, geometric, or neither

d = 5, arithmetic

  1. Find the # of terms

an = 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 + an)/2

= 10(7 + 52)/2

= 295

64
Q

sigma notation

A

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

66
Q

expanded form

A

when you substitute the values of n into the formula

68
Q

geometric series

A

its terms form a geometric sequence

70
Q

sum finite geometric series

A

S =

a1 - anr

________

r

72
Q

evaluating sigma notation

A

​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
74
Q

partial sum

A

the sum of the first n terms of an infinite series

76
Q

sequence of partial sums

A

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

78
Q

sum of an infinite series

A

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

80
Q

finding sums of infinite series by graphing

A

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
82
Q

sum infinite geometric series

A

for any infinite geometric series with |r| < 1, the terms get closer & closer to 0. This suggests substituting 0 for an in the formula for the sum of a finite geometric series

S =

a1 - anr

_______

1 - r

►

a1 - 0 • r

________

1 - r

►

a1/1 -r for |r| < 1

84
Q

finding sums of infinite series by formula

A

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

Math II (17 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions