sequence

an ordered list of numbers

terms

the numbers in a sequence

finite sequence

a sequence with a last term

infinite sequence

a sequence with no last term

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

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

explicit formula

gives the value of any term a_{n} in terms of n

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. ||

a_{1} = 90

a_{2 }= 83 = 90 - 1(7)

a_{3} = 76 = 90 - 2(7)

a_{4} = 69 = 90 - 3(7)

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

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 v_{0 }for initial deposit. Next term = v_{1}, for 1st month's interest, etc. etc.

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. d_{0} = 10,000

d_{1} = 10,000(1.1)

d_{2} = d_{1}(1.1)

= 10,000(1.1)(1.1)

= 10,000(1.1)^{2}

d_{3} = d_{2}(1.1)

= 10,000(1.1)^{2}(1.1)

= 10,000(1.1)^{3}

•

•

•

d_{n} = 10,000(1.1)^{n}

basically, x(1 + rate)^{n}

fractal

this process continues without end

self-similar

can be used to make a recursive formula

self-similarity

the appearance of any part = similar to the whole thing

recursive formula

tells how to find the *n*th term from the term(s) before it.

2 parts:

- a
_{1}= 1 ⇔ value(s) of 1st term(s) r given - a
_{n}= 2a_{n-1 }⇔ recursion equation

recursion equation

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

finding recursive formulas

Ex: 1, 2, 6, 24

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

a_{1} = 1

a_{2} = 2 = 2 • 1

a_{3} = 6 = 3 • 2

a_{4} = 24 = 4 • 6

- Write in terms of a

a_{2} = 2 • a_{1}

a_{3} = 3 • a_{2}

a_{4} = 4 • a_{3}

- Write a recursin equation

a_{n} = na_{n-1}

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

a_{1} = 1

a_{n} = na_{n-1}

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?

- Write a recursion equation

amount aspirin after *n*th dose = 26% amount after prev. dose + new dose of 650mg

a_{n} = (0.26)(a_{n-1}) + 650

- Use a calculator

Enter a_{1} → 650

Enter recursion equation using ANS for a_{n-1} → .26ANS + 650

Keep pressing Enter

**sequence appears to approach limit of about 878**