Lesson # 18: Infinite Sum & Sigma Sum

Infinite Sum:

Add the infinite sum of 6+2+2/3+.....

S =     6     
     (1-(-1/3))
   =      6    
         (2/3)
   = 6(3/2)
   =9

What is the formula:  S=   a   
                                       l-r

Sigma:

5

∑ 3i+2
i=1

5 - the terms
i =1  - input #
3i+2 - formula

Lesson # 17: Geometric Sequences & Series

Formula:

tn = ar^n-1
tn - value
n - the position
r - multiplier (2nd/1st)

Example:             2, 6, 18, 54,......
1) t10
tn = 2(3)^9
tn = 39 366

2) How many terms are in the following sequence?   8, 12, 18, 27,...... 205.03125
205.03125 = 8(1.5)^n-1
        8                  8
log(25.6289) = (n-1) log(1.5)
   log(1.5)             log(1.5)
8 = n-1     
9 = n


Series:
 A series is the sum of a sequence
S= add   t= find number

Formula:
 
Sn = a(1 - r^n)   or   Sn = a- rl                           
             1-r                       1-r      
 
Sn: series (sum)       
r: ratio     
l: last term> tn
a: first #         
n: how many terms there is

Example:
1) Find the sum of the series          5 + 15+ 45 +.......... + 885 735
Sn =  a - rl
          1-r               
Sn= 5 - 3(885 735)
             1-3                                                                                      
     =  1, 328, 600

Lesson # 16: Richter Scale, pH Scale & Decibels

Earth Quake: Richter Scale




4.0 ------x10------ 5.0 ------x10 ------ 6.0 ------x10 ------ 7.0


R = 10    P = 1

pH Scale:

Sounds- decibels dbs:

40 dbs  > whisper      90 dbs > shout

           |  100000 times louder   |

R= 10      P= 10


Example 1:
How much more intense is an Earthquake of 6.8 to a 3.5?
F = IR ^ t/p                     M= magnitude   M=F       I= 1            t= change   
M= 10^ 3.3/1
M= 1995.26

Example 2:
A chainsaw of 105dbs to a whisper of 40dbs
M = R^t/p
M= 10^ 65/10                     
M= 10^ 6.5
M= 3162277.66

Example 3:
Find the pH that is 80x more basic than water?
water= pH 7
M = R ^t/p
80= 10 ^t
log 80 = t log10                    *note log10 = 1*
1.90= t
7+ 1.9= 8.9
t= 8.9

Lesson # 15: Exponential Growth and Decay

There are two different formulas:

The most common formula to use is:

F= IR^ t/p                    F: final amount       I: initial amount     R: rate of growth ( 5% = 0.05 +1 R>1)   growth decay ( 5%  R= 0.95)     t: time       p: time for R to occur (days, weeks, months,etc)  

* note: t and p must be in same units

The second formula is for continuous interest rates:

 P= Po e^ Kt          P: final amount       Po: initial amount      e: calculator function   K: growth/decay (no 1)  

t: time 

Example 1:

What will $3500 grow to, if invested at 6% interest for 10 years, compounded annually?

F= IR ^ t/p               F=?     I= 3500   R= 1.06  t= 10    p=1 

(*note: if p is a fraction take the reciprocal and multiply it to the top number) 

F= 3500(1.06)^10/1

F= 3500(1.06)^10

F= $6267.97

Example 2:

What amount of money would grow to $ 4000 if invested at 91/4%, compounded annually for 4 years?

F= IR ^ t/p              F= 4000   I= ?   R= 1.0925   t= 4     p=1

4000   =     I  (1.0925) ^ 4/ 1

(1.0925)^4     (1.0925)^4

I= $2807.85

Example 3:

In April. the atmosphere at Chernobyl was contaminated with radioactive iodine-131, which has a half-life of 8.1 days.  How long did it take for the level of radiation to reduce to 1% of the level immediately after the accident?

*Half-life  R= 0.5     P= half-life
1/100 = 100(0.5)^t/8.1
0.01 = 0.5^t/8.1
log0.01 = t/8.1 log0.5
8.1 log0.01/log0.5 = t