Binomial Theorem

Binomial= a polygon that has two terms.

Expanding a Binomial:

(x + y)¹ = 1x + 1y

(x + y)² = 1x² + 2xy + 1y²

(x + y)³ = 1x³ + 3x²y + 3xy² +1y³

(x + y)⁴ = 1x⁴ + 4x³y + 6x²y² + 4xy³ + 1y⁴

In each expansion there is (n + 1) terms

n= 2 → (2+1) = 3 terms

n= 3 → (3+1) = 4 terms

The power of each term adds up to the power each binomial is added to (n).

For, (x + y)⁴ = 1x⁴ + 4x³y¹ + 6x²y² + 4x¹y³ + 1y⁴

Powers: (4) (4) (3+1= 4) (2+2=4) (1+3=4) (4)

where n = 4, the sum of powers are 4.

Coefficients of the expansion correspond to the Pascal's Triangle

*During expansion, the x and y have symmetric roles. As the powers of "x" decrease by 1, the powers of "y" increase by 1.

(x + y)⁴ = x⁴(y⁰) + 4x³y¹ + 6x²y² + 4x¹y³ + (x⁰)y⁴

To find the coefficents you can also use the formula:

_nC_{k =}

The 2 values next to each others' sum is equal to the number directly below the values. This is the case for all the numbers on Pascal's triangle.

Geometric Sequences and Series

A Geometric Sequence is the ratios of consecutive terms that are the same.

the number r is the common ratios of the sequence.

The difference between an Arithmaic Sequence and Geometric Sequence

Arithmatic Geometric

Geometric

The sequence whose nth term is 2^n is geometric. For this seqence, the common ratio between consecutive terms is 2.

2,4,8,16...2^n...

4/2= 2

The sum of Geometric Sequences

if r is less that absolute value 1 then the equation is s=a1/1-r

Need more help, follow the link

http://www.youtb.com/watch?v=IGFQXInm-co

soft



lv

Arithmetic Sequences and Partial Sums

Definition of an Arithmetic Sequence - a sequence whose consecutive terms have a common difference

The common difference (d) is the difference between two consecutive numbers in the sequence.

Example:

4, 8, 11, 14, 17,...

d=3

You can find the nth term of an Arithmetic Sequence by using the equation...

an = dn + c

where c = a1 -d

The equation can also be written as an = d(n-1) + a1

Example:

2, 6, 10, 14, 18

d=4

c=2

an = 4n - 2

The equation for the Sum of a Finite Arithmetic Sequence...

Sn = (n/2)(a1 + an)

You can use this equation to add up the numbers in a Arithmetic Sequence.

Example:

Sn = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 +.....+ 98 + 99 + 100

= (100/2)(1 + 100)
= 50(101)
=5050

You can find a Partial Sum of an Arithmetic Sequence by using the equation for the Sum of a Finite Arithmetic Sequence.

Example:

5, 16, 27, 38, 49,....
Find the 150th term

an = dn + c
a150 = 11(150) - 6
a150 = 1644

S150 = (150/2)(5 + 1644)
S150 = 75(1649)
S150 = 123,675

Sequence and Series

Definition of Sequence- An infinite sequence is a function whose domain is the set of positive integers. The funciton values

are the terms of the sequence. If the domain of the functioning consists of the first n positive integers only, the sequence is a finite sequence.


Finding the Terms of a sequence





The Fibonacci Sequence: A Recursive Sequence

Definition of Factorials- If n factorial is defined by

As a special case, zero factorial is defined as 0! = 1.

• 0!=1



• 1!=1



• 2!=1 x 2 = 2



• 3!= 1 x 2 x 3 = 6
  
  
  

Evaluating Factorial Expressions

Definition of Summation Notation- The sum of the first n terms of a sequence is represented by

Where i is called the index of summation, n is the upper limit of summation, and 1 is lower the limit of summation

Sigma Notation for Sums

Systems of Equations

hi.

there are three ways we know how to solve systems of equations

substitution
{x+y = 10, x-y = 4} {equation 1, equation 2}

x=10-y {solve for x in equation 1}

(10-y)-y=4 {substitute 'solution for x in equation 1' into equation 2}
10-2y=4 {simplify}
2y=6 {simplify}
y=3 {solve for y}
x+(3)=10 {plug y into equation 1}
x=7 {solve for x}

solution: (7, 3)



elimination

{x+y = 10, x-y = 4} {equation 1, equation 2}

{x+y=10 {add (or subtract) equations, eliminating one variable}
{x-y=4
-2y=-6 {determine new equation}

y=3 {solve for y}
x+(3)=10 {plug y into equation 1}
x=7 {solve for x}

solution: (7, 3)


graphing

{x+y = 10, x-y = 4} {equation 1, equation 2}

plug both into calculator

find where the graphs intersect each other

solution: (7, 3)


fun stuff i know

bye.

Ways to Solve Logarithmic and Exponential Equations

The new way we learned to solve logarithmic equations is exponentiation. Exponentiation acts as an inverse sine, or a square root to make it easier to solve an equation. Just like a square root takes away certain powers, so does exponentiation.

ex.

ex. x=2 b=10 a=100

Exponential Equations

One -to-One Proerties
a^x=a^y if and only if x=y
Logx=Logy if and only if x=y
Example
2^x=32
2^x=2^5
x=5

Inverse Properties
a^logx=x
loga^x=x
Example
e^x=7
lne^x=ln7
x=ln7

Strategies for Solving Exonential and Logarithmic Equations
1. Rewrite the given equation in a form to use the One-to-One properties of exponential or logarithmic functions.
2. Rewrite an exponential equation inlogarithmic form and applly the Inverse property of logarithmic functions.
3. Rewrite a logarithmic equation in exponential formand apply the Inverse Property of exponential functions.

Example
Solve 2(3^(2t-5))-4=11
2(3^(2t-5))-4=11 Write original equation
2(3^(2t-5))=15 Add 4 to each side
3^(2t-5)=15/2 Divide each side by 2
log'3 3^(2t-5)=log'3(15/2) Take log base 3 of each side
2t-5=log'3(15/2) Inverse Property
2t=5+log'3(7.5) Add 5 to each side
t=(5/2)+(1/2)log'3(7.5) Divide each side by 2
t~3.417 Use a calculator