Wednesday, April 17, 2013

01 Basics of Arithmetic & Geometric Progression (Grades 9 to 12) Part-01

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Dear Students,

Download following files for learning Formulas more effectively. Take the print out of these files and write your answers daily to improve your scores in 10th standard/grade.


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1) Algebra Formulas
2) Geometry Formulas

In any Music, March past or any other rhythmic work we observe that these tasks are done with  proper sequence. Let us take a very simple example of our day today work. In night we sleep, then in the morning we wake up, then we finish our morning tasks. We take our breakfast then go to school/office/business-spot/work-place. We take lunch in the afternoon outside those who go out else at home/hotel. Then we come home in the evening then do some entertainment, watch TV or play some computer Games then diner then go to bed to sleep. This cycle will be maintained on week days. This one may be an example of sequence if we say that each and every task is done very well.  

Harmonic Motion, Sewing needle, Swing of Pendulum are also the examples of sequence.

1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2....... is also an example of sequence. here the three numbers 1, 3, 2 are getting repeated several times.

Another example will be 1, -1, 1, -1, 1,1, -1, 1, -1, 1,1, -1, 1, -1, 1,1, -1, 1, -1, 1....... 
(The position of the Sewing Needle)



In Math, we will study three types of sequences. 

1) Arithmetic Sequence (Arithmetic Progression) Written as A. P.
2) Geometric Sequence (Geometric Progression) Written as G. P.
3) Harmonic Sequence (Harmonic Progression) Written as H. P.

Details of Arithmetic Sequence (Arithmetic Progression):

When a constant Number is added in the previous term we get our term and the same procedure is done for all new succeeding term is called Arithmetic Sequence or Arithmetic Progression.

Each number of the sequence is called as term and their position is called as its place such as first term, second term, third term, .... r'th term (this is the general term),.... n'th term and so on.

Generally sequence is denoted by <an> , (an) , <tn>; or (tn) and n term is denoted as tn.
In Arithmetic Progression, the difference between two succeeding terms is known as " Common Difference " and is denoted by " d "

In Arithmetic Progression, the first term id denoted by " a ".

See the following table to understand the concept of Arithmetic Progression. (It will also help us to understand the method of finding the nth term of an Arithmetic Progression.

Places
1st Place
2nd  Place
3rd Place
---
rth Place
---
---
nth Place
Notations
t1
t2
t3

tr


tn
Terms
a
a + d
a + 2d

a + (r-1) d


a + (n-1)d

Second Method to get nth term of an Arithmetic Progression.

By definition of the common difference of an Arithmetic Progression, we have,

   (t2  - t1)    = d    [Note: 1st   " d ", which is suffix number of the 2nd  part of (t2  - t1) ]
   (t3  - t2)    = d    [Note: 2nd  " d ", which is suffix number of the 2nd  part of (t3  - t2) ]
   (t4  - t3)    = d    [Note: 3rd   " d ", which is suffix number of the 2nd  part of (t4  - t3) ]
------------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------------
   (tn  - tn-1) = d    [Note: (n-1)th  " d ", which is suffix number of the 2nd  part of (tn  - tn-1) ]
------------------------------------------------------------------------------------------------------------------------
   tn  - t1   =  (n-1) d    [ addition of all above terms ]

   tn          =  t1 + (n-1) d  
   tn          =  a + (n-1) d 
Summation of an AP:

Before discussing about the formula of sum of first n terms of an AP, let us see one interesting example .

Find the sum of first 100 Natural Numbers.


Let us denote the sum of first 100 natural Numbers by  " S100 "

     S100 = 001 + 002 + 003 + --------- + 099 + 100
     S100 = 100 + 099 + 098 + --------- + 002 + 001
 --------------------------------------------------------------------
  2 S100 = 101 + 101 101 + --------- + 101 101 
  2 S100 = 100 * 101 
    S100 =  (100 * 101)/2
    S100 = 50 * 101
    S100 = 5050

By applying the same technique, we will find the sum of first n terms of an AP.

Let our AP has its first term as "a" and Common difference as "d". Let us denote the sum of first n terms of an AP be Sn. Here we will take the last term as " l "  so here l  =  a + ( n - 1 ) d

     S=   a +  (a + d) + (a + 2 d) + --------- + (l - d) + l

     S=   l   +  (l - d)   (l - 2 d) + --------- + (a + d)+ a
 --------------------------------------------------------------------
   2 S= (a+l) + (a+l) (a+l) + --------- + (a+l) + (a+l)
   Sn = n * (a+l)
      Sn = n * (a+l)/2
      Sn = n * [ a + ( a + (n-1) d ) ] / 2
      Sn = n * [ 2 a + (n-1) d ) ] / 2

Some examples of this session will be discussed in the next Blog.

Anil Satpute