Tid Bits from Chapter 2

Deriving the Quadratic Formula: 
Solve ax² + bx + c = 0 by completing the square as follows:
1. Subtract c from both sides
ax² + bx = -c
2. Divide both sides by a, the leading coefficient

x² + b/a x = -c/a

3. Add the square of half of b/a, that is, (b/2a)², to both sides

x² + b/a x + (b/2a)²= -c/a (b/2a)²

4. Write the left side of the equation as a perfect square

(x + b/2a)² = (b/2a)²-c/a

5. Take the square root of both sides

x + b/2a = √(b/2a)²-c/a

6. Subtract b/2a from both sides

x = -b/2a + or - √(b²-4ac)/(2a)

Quadratic Formula:

x = -b/2a + or - √(b²-4ac)/(2a)

Summation

Summation Notation: Finds the sum of various terms in a sequence. 

This diagram is the summation sign.

The Equation for Summation Notation:

m

∑ ck means C₁+C₂+C₃+….+Cm

k=1

EXAMPLE:

5

∑ (-7+3n) 

n=1

=(-7+3x1) + (-7+3x2) + (-7+3x3) + (-7+3x4) + (-7+3x5)

=(-7+3) + (-7+6) + (-7+9) + (-7+12) + (-7+15) 

=-4-1+2+5+8

=10

Partial Sum: The sum of the first k terms of the sequence.

Formulas for the Partial Sum:

1.

 k

∑ un = k/2 (u₁ + uk)

n=1

2.

k

∑ un = ku₁ + k(k-1)/2  x d

n=1

Using the Calculator to do Partial Sums:

step 1: Press 2nd STAT, highlight MATH and scroll down to number 5: sum( , hit enter

step 2: Press 2nd STAT, highlight OPS and scroll down to number 5: seq( , hit enter

step 3: enter your equation!

EXAMPLE:

A ball is dropped from a hight of 5 feet. It hits the ground and bounces to a hight of 41/2 feet. It continues to bounce, and on each rebound it rises to 9/10 the hight of the previous bounce. 

1. What is the total distance the ball travels through the fifteenth bounce?

       k

5 + ∑ (9) (.9)(n-1)

      n=1

2. What is the command line on your calculator used to calculate the total distance the ball traveled through the fifteenth bounce?

5 + sum(seq(9(.9)(n-1),n,1,14)

Geometric Sequences

Geometric Sequence: A sequence in which terms are found by multiplying a preceding term by a nonzero constant.


Recursive Formula: U_n=run-1
The recursive formula, for geometric sequences, uses the terms, preceding the term you are trying to find, to find the term you need to find.

Explicit Formula: U_n = U₁r^{(n - 1)}

The Explicit formula, for geometric sequences, allows you to find a term within the sequence based on the position in the sequence the term is in.


EXAMPLE:
In a geometric sequence, if U₃=125 and U₅=3125
a. What is the constant ratio?
U₅/ U₃ = 3125/125
U₁r²/U₁r⁴ = 3125/125 = √r² = √25
r = 5
b. Find U₈.

U₈ = (5)5⁽⁸⁻¹⁾ = 390,625

Arithmetic Sequences

Arithmetic Sequences: A sequence in which the difference between each term and the preceding term is always constant.

The Explicit Formula: an = a1 + (n – 1)d An explicit formula, for arithmetic sequences, allows you to find a term within the sequence based on the position in the sequence the term is in. The Recursive Formula: The recursive formula, for arithmetic sequences, allows you to find a term by using the preceding terms in the sequence.

EXAMPLE: 
The First term, U₁, and the common ratio, r, of a geometric sequence are given. Find the sixth term and the recursive and explicit formulas for the nth term

U₁ = 1, r = -2
Sixth Term: U₆ = 1(-2)⁵---U₆ = -32

Recursive Form:
U₁=1

U₂=(-2)1=-2

U₃=(-2)(-2)=4

U₄=(4)(-2)=8

U₅=(-8)(-2)=16

U₆=(16)(-2)=-32

Explicit Form:

U₁=1

U₂=(1)(-2)=-2

U₃=(1)(-2)²=4

U₄=(1)(-2)³=-8

U₅=(1)(-2)⁴=16

U₆=(1)(-2)⁵=-32