Geometric Series and Binomial Expansions

I.  Homework

II.  Geometric Sequences

Find the general term of the following:

A)  1,2,4,8,16,...

B)  27,9,3,1,1/3,...

C)  3,6,12,24,48,...

D)  1/2,-1,2,-4,8,...

A Geometric sequence is a sequence which the ratio of the common terms is equal.

The general term is

a_n = a₁r^n-1 where r is the common ratio.

Example:  Find the general term of the geometric series such that

a₅ = 48 and a₇ = 192

Solution:  We have that a_n = a₁rⁿ  which gives the equations

48 = a₁r⁴ , 192 = a₁r⁶

Dividing the two equations, we get:    

4 = r²

Hence r = 2 (or r = -2)

Substituting back into the first equation, we get

48 = 16a₁

So that a₁ = 3

Hence the general term of the sequence is

a_n = (3)(2)^n-1  or a_n = (3)(-2)^n-1  

III.  Geometric Series

Theorem:  S_n = a₁(1 - rⁿ)/(1 - r)

Example:  Find the sum

5 + 10 + 20 + 40 + ... + 2560

Solution:     a₁ = 5 and r = 2 and n = 10 so that

S_n = a₁(1 - rⁿ)/(1 - r) = 5(1 - 2¹⁰)/(1 - 2)

For an infinite geometric series if  |r| <1 then

S_infinity   =   a₁/(1 - r)

Example:  How much is going to taxes?  Suppose that we track $100 each time money is spent 8% goes towards taxes and the rest gets respent.  How much of the $100 is spent before it is all taxed out?

  a₁ = 100, r = .08,

S_infinity   =   a₁/(1 - r) = 100/(1 - .08) =100/.02 = $5,000

IV.  Binomial Expansions

The class will show that

(x + y)⁰ = 1 

(x + y)² = x² + 2xy + y²

(x + y)³ = x³ + 3x³y + 3xy² + y³  

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

Notice that the powers are descending in x and ascending in y.  

V.  Pascal's Triangle

We can write the coefficients suggestively as

      1

    1   1

   1  2  1

  1 3  3  1

1  4  6  4  1  

We see that a number is constructed by adding the two numbers above it.  The borders of this triangle are all ones.

The class will write the next rows of the triangle.  

We use this triangle as follows:

Method of determining (x + y)ⁿ  

Example:  (x + y)⁵  

1)  Write the nth row of Pascal's Triangle leaving lots of space

1    5    10    10    5    1

2)  Start with xⁿ  and insert it next to the first term of the written row of Pascal's Triangle.   Add a + sign.

1x⁵  + 5    10    10    5    1

3)  insert x^n-1y next to the second number of Pascal's Triangle and add a + sign.

1x⁵  + 5 x⁴y + 10    10    5    1

4)  Continue this process decrementing the power of x and incrementing the power of y until you place the term yⁿ  next to the final number.

1x⁵  + 5 x⁴y + 10x³y² + 10x²y³ + 5xy⁴ + 1y⁴  

Exercises:  Expand

A.  (2x - y)⁴

B.  (x - y)⁶

VI)  Factorials and Pascal's Triangle

We define 5! = (5)(4)(3)(2)

and n! = (n)(n - 1)(n - 2)....(3)(2)

Theorem:  In the expansion of (x + y)ⁿ  the coefficient in front of the term

x^n-ky^k   

is n!/[(n - k)!(k!)]

Example:

In the expansion (x + y)⁹ the coefficient in front of x³y⁶ is

9!/[(9 - 6)!(6!)] = 9(8)(7)(6)(5)(4)(3)(2)/(3)(2)(6)(5)(4)(3)(2)

= 9(8)(7)/(3)(2) = 3(4)(7) = 84

We will do more examples in class.    

A.  Find the 8th term of the  expansion of the binomial

(2x - y)¹⁴

B.  Find the third term of expansion of the binomial

(x + 3y)¹⁰