## Wednesday, April 17, 2013

### 51-01 Basics of Arithmetic & Geometric Progression (Grades 9 to 12) Part-01

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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 to day work. At 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 dinner then go to bed to sleep. This cycle will be maintained on weekdays. This one may be an example of a 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 a 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.......

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 a term and their position is called as its place such as first term, second term, third term, .... rth term (this is the general term),.... nth 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 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

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) ]
tn          =  t1 + (n-1) d

Before discussing 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 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

(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.

In Arithmetic Progression, the difference between two succeeding terms is known as " Common Difference " and is denoted by " d "

Second Method to get nth term of an Arithmetic Progression.
(t2  - t1)    = d
(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:

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.

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.