4.1-4.3 Flashcards Preview

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Flashcards in 4.1-4.3 Deck (16)
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1

sequence

an ordered list of numbers

2

terms

the numbers in a sequence

3

finite sequence

a sequence with a last term

4

infinite sequence

a sequence with no last term

5

graphing sequences

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

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

6

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

7

explicit formula

gives the value of any term an in terms of n

8

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

          a= 83 = 90 - 1(7)

          a3 = 76 = 90 - 2(7)

          a4 = 69 = 90 - 3(7)

3. Express the pattern in terms of n. || an = 90 - (n - 1)7

 

 

9

subscript 0

When the 1st term of a sequence represent a starting value before any change occurs, subscript 0 is often used.

Ex: monthly bank account balances, 1st term is vfor initial deposit. Next term = v1, for 1st month's interest, etc. etc.

10

percentage explicit formulas

Ex: bacteria count increaes 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)

2. 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

11

fractal

this process continues without end

self-similar

can be used to make a recursive formula

12

self-similarity

the appearance of any part = similar to the whole thing

 

13

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) r given
  2. an = 2an-1      ⇔ recursion equation

14

recursion equation

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

15

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 recursin equation

an = nan-1

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

a1 = 1

an = nan-1

16

percentage recursive formulas

Ex: 650mg of aspirin every 6h; only 26% of aspiring 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