Ch. 4 Flashcards
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 = a1- anr / 1-r
sum geometric infinite series → S = a1 / 1-r; |r| < 1
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), u 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 an in terms of n
finding explicit formulas
Ex: Given 90, 83, 76, 69,…, find a formula for the sequence
- Find a pattern. || a repeated subtraction of 7
- 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)
- Express the pattern in terms of n. ||
an = 90 - (n - 1)7
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.
percentage explicit formulas
Ex: bacteria count increases 10% each day; 10,000 now; Find formula for bacteria count after n days
- d + .1d {if annual interest, compounded monthly, then d + (.1/12)d}
d(1 +.1)
d(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
self-similarity
the appearance of any part is similar to the whole thing
recursive formula
tells how to find the nth term from the term(s) before it
2 parts:
- a1 = 1 ⇔ value(s) of 1st term(s) given
- an = 2an-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
a1 = 1
a2 = 2 = 2 • 1
a3 = 6 = 3 • 2
a4 = 24 = 4 • 6
- Write in terms of a
a2 = 2 • a1
a3 = 3 • a2
a4 = 4 • a3
- Write a recursion equation
an = nan-1
- Use value of 1st term & recursion equation to write recursive formula
a1 = 1
an = nan-1
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?
- Write a recursion equation
amount aspirin after nth dose = 26% amount after prev. dose + new dose of 650mg
an = (0.26)(an-1) + 650
- 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
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
common difference
d
the constant value an - an-1
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
common ratio
r
the constant value an/an-1
explicit arithmetic formula
an = a1 + (n - 1)d
recursive arithmetic formula
a1 = value of 1st term
an = an-1 + d
using explicit formulas to find out how many terms are in a finite sequence
Ex: 33, 29, 25, 21,…, 1
- arithmetic, geometric, or neither?
d = -4 → arithmetic
- 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
explicit geometric formula
an = a1rn-1
recursive geometric formula
a1 = value of 1st term
an = (an-1)d
geometric mean
sqrt(ab)
Ex: find x in geo sequence 3, x, 18
- 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)
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
finite series
has a last term
infinite series
has no last term
arithmetic series
its terms form an arithmetic sequence
sum finite arithmetic series
S =
n(a1 + an)
_________
2
finding the sum of a finite series
Ex: 7 + 12 + 17 + 22 +…+ 52
- arithmetic, geometric, or neither
d = 5, arithmetic
- Find the # of terms
an = a1 + (n -1)d
52 = 7 + (n - 1)5
52 = 7 + 5n - 5
50 = 5n
10 = n 10 terms
- Use formula
S = n(a1 + an)/2
= 10(7 + 52)/2
= 295
sigma notation
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
expanded form
when you substitute the values of n into the formula
geometric series
its terms form a geometric sequence
sum finite geometric series
S =
a1 - anr
________
r
evaluating sigma notation
if finite,
- write in expanded form
- decide if arithmetic, geometric, or neither
- use formula for sum of whatever type it is
if infinite,
- see whether a sum even exists first by expanding first few terms
- if geometric, use ratio to see if sum exists. if arithmetic, graph.
- find sum using formula or graphing
partial sum
the sum of the first n terms of an infinite series
sequence of partial sums
2, (2 + 4), (2 + 4 + 6), … forms a sequence of partial sums for the infinite series 2 + 4 + 6 +…
sum of an infinite series
if the sequence of the partial sums of an infinite series has a limit, then that limit is the sum of the series
finding sums of infinite series by graphing
Ex: 3 + (-1.5) + .75 + (-0.375) + …
- Find common ratio → r = 0.5
- 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
- Graph the partial sums (position, term)
- limit is about 2, so sum is about 2
sum infinite geometric series
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
finding sums of infinite series by formula
Ex: 4 + 4/5 + 4/25 + 4/125 + …
- must be geometric with |r| < 1
geometric; r = 1/5
- Use formula for infintie geometric series
S = a1/1-r = 4/(1-1/5) = 5
- the sum of the series is 5
