4.1-4.3

Question 1

sequence

Answer 1

an ordered list of numbers

Question 2

terms

Answer 2

the numbers in a sequence

Question 3

finite sequence

Answer 3

a sequence with a last term

Question 4

infinite sequence

Answer 4

a sequence with no last term

Question 5

graphing sequences

Answer 5

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

Question 6

limit of a sequence

Answer 6

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 7

explicit formula

Answer 7

gives the value of any term a_n in terms of n

Question 8

finding explicit formulas

Answer 8

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. ||

      a<sub>1</sub> = 90
   
      a<sub>2</sub>= 83 = 90 - 1(7)
   
      a<sub>3</sub> = 76 = 90 - 2(7)
   
      a<sub>4</sub> = 69 = 90 - 3(7)

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

Question 9

subscript 0

Answer 9

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 v₀for initial deposit. Next term = v₁, for 1st month’s interest, etc. etc.

Question 10

percentage explicit formulas

Answer 10

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)dd(1 + .1)d(1.1)
2. d₀ = 10,000d₁ = 10,000(1.1)d₂ = d₁(1.1)

     = 10,000(1.1)(1.1)
   
     = 10,000(1.1)<sup>2</sup>

   d₃ = d₂(1.1)

     = 10,000(1.1)<sup>2</sup>(1.1)
   
     = 10,000(1.1)<sup>3</sup>
   
            •
   
            •
   
            •

   d_n = 10,000(1.1)ⁿbasically, x(1 + rate)ⁿ

Question 11

fractal

Answer 11

this process continues without end

self-similar

can be used to make a recursive formula

Question 12

self-similarity

Answer 12

the appearance of any part = similar to the whole thing

Question 13

recursive formula

Answer 13

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. a_n = 2a_n-1 ⇔ recursion equation

Question 14

recursion equation

Answer 14

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

Question 15

finding recursive formulas

Answer 15

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

a_n = na_n-1

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

a₁ = 1

a_n = na_n-1

Question 16

percentage recursive formulas

Answer 16

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

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