51-01 Basics of Arithmetic & Geometric Progression

In any Music, March past, or any other rhythmic work, we observe that these tasks are done in 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/workplace. 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, and Swing of the Pendulum are also 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, and 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 to the previous term we get our term and the same procedure is done for all new succeeding terms is called Arithmetic Sequence or Arithmetic Progression.

Each number of the sequence is called a term and its position is called 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, the sequence is denoted by <a_n>, (a_n), <t_n>; or (t_n) and n term is denoted as t_n.

In Arithmetic Progression, the first term is 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} n^th _{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: 1^st   " d ", which is suffix number of the 2^nd  part of (t₂  - t₁) ]

   t_{n          =}  t_1 + (n-1) d  

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

Find the sum of the first 100 Natural Numbers.

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

     S_n =   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 "}

The second method is to get n^th term of an Arithmetic Progression.

   (t₂  - t₁)    = d

   (t₃  - t₂)    = d    

                [Note: 2^nd  " d ", which is the suffix number of the 2^nd  part of (t₃  - t₂)]

   (t₄  - t₃)    = d    

                [Note: 3^rd   " d ", which is the suffix number of the 2^nd  part of (t₄  - t₃)]

   (t_n  - t_n-1) = d    

         [Note: (n-1)^th  " d ", which is the suffix number of the 2^nd  part of (t_n  - t_n-1)]

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   t_n -  t_{1   =  (n-1) d    [ addition of all above terms ]}

   t_{n          =}  t_1 + (n-1) d

   t_{n          =  a +} (n-1) d  

Summation of an AP:

Let us denote the sum of the 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 the first n terms of an AP.

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

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   2 S_n = (a+l) + (a+l) + (a+l) + --------- + (a+l) + (a+l)

   2 S_n = n * (a+l)

      S_n = n * (a+l)/2

      S_n = n * [ a + ( a + (n-1) d ) ] / 2

      S_n = n * [ 2 a + (n-1) d ) ] / 2

In the next part, we will see a few examples and some essential formulae.

Click here for the next basics.

Anil Satpute