Int Alg Section 8.6 Day 2
Counting Principles
1) Does the problem involve 2 or more separate events?
Ifyes.FundamentalCountingPrinciple
2)Doesordermatter?Ifyes.Permutation
3) Are the chosen elements a subset of a larger set where order is not
important?If
Chapter 7 Quiz Review
Solve for x and y.
1. [
]
2. 2. [
In 3-10, given A = [
[
]
]
[
]
] and B = [
3. A B
4. B A
6. 4A + B
7.
], find:
5.
8.
9. A + B + A
10. A + B A
Multiply
11.
12.
[
][
[ ][
13. Find the inverse of [
]
]
]
14. Find the inverse of [
]
15
Int Alg Section 8.5 The Binomial Thm
Binomial Coefficients
There are several observations you can make about these
expansions.
1. In each expansion, there are n + 1 terms.
2. In each expansion, x and y have symmetrical roles. The
powers of x decrease by 1
8.3 Geometric Sequences and Series
geometric sequence
r = a2 a1, a3 a2, an+1 an (next term preceding term)
_ _
an = a1 rn-1
Explicit form of a geometric sequence
Ex 1: Determine whether each sequence could be geometric or arithmetic. If possible,
find the
Int Alg 8.6 Counting Principals
Probability has to do with counting the number of ways an event can occur.
Sometimes the easiest way to get that number is to simply write down or count every
possibility.
Ex) The numbers from 1 to 5 are written on pieces o
8.4 Rules for Sums of Powers of Integers
n
1)1+2+3+4+nor
k=
k 1
n
2) 12 + 22 + 32 +n2 or
n(n 1)
2
k2 =
k 1
k3 =
n 2 (n 1) 2
4
k4 =
n(n 1)(2n 1)(3n 2
30
k5 =
n
3) 13 + 23 + 33 +.+n3 or
n 2 (n 1) 2 (2n 2
12
k 1
n
4) 14 + 24 + 34 +.+n4 or
k 1
n
5) 15 + 25 +
The explicit formula can be generalized to:
ay = ax
ry-x
Notice an = a1rn-1 so a7 = a3r7-3.
Ex 3: Find the 8th term of the geometric sequence with a3 = 36 and a5 = 324.
Ex 4: Find the 7th term of the geometric sequence with a4 = -8 and a5 = -40.
Day 1: Do
8.1 Day 2
Writingexplicitexpressionsforsequencesandseries.
Going backwards from what we did yesterday now we are given a list of terms
and we come up with the rule.
The following are 4 common types of sequences.
1) Sequences with a common difference (Arit
Series: Sigma = The summation symbol
6
Ex)
. Sigma means to find the sum.
(3n 5)
n 1
5
Ex)
k2
k=0
Assignment
8.1.2/ 48-60 evens, 67-70, 86-92 evens.
8.2 Arithmetic Sequences and Partial sums
A sequence is arithmetic if it has a common difference. (a2 a1 =
8.1 Sequences and Series
Sequence: An ordered set of terms.
Series: A sum of all or some of the terms of a sequence.
Examples:
5,10,15,20,25,30,35,
1+3+5+7+9+.
3 + 4 5 + 6 7 + 8 9+
1,3,9,27,81,
Sequences can be defined either explicitly or recursively.
Ex
8.5 Day 2 Notes
(x +y)n expansion
nCrx
n-r r
y for the (r + 1)th term
Ex) Find the 4th term of (x + y)10 so n = 10
Ex) Find the 8th term of (4x + 3y)9
Binomial Probability
k n-k
nCkp q
is the probability of k successes in n trials. P = probability of succ
Intermediate Algebra Chapter 8 Formula Sheet
n
k =
1) 1+2+3+4+ + n or
k 1
2) 12 + 22 + 32 + + n2 or
n(n 1)
2
n
k =
2
k 1
3
3
3
3
3) 1 + 2 + 3 + . + n or
n
k =
3
k 1
4) 14 + 24 + 34 + . + n4 or
n
k4
=
k 1
5) 15 + 25 + 35 + . + n5 or
n
k5 =
k 1
Explicit for