Arithmetic Progressions

 

 

An arithmetic progression is a sequence in which each term (except the first term) is obtained from the previous term by adding a constant known as the common difference. An arithmetic series is formed by the addition of the terms in an arithmetic progression.

 

Let the first term on an A. P. be a and common difference d. Then,

 

General form of an A. P.:

            a, a + d, a + 2d, a + 3d, ...

 

nth term of an A. P.:

            a + (n - 1) d

 

Sum of first n terms of an A. P.:

            n/2 [2a + (n - 1) d]    or

            n/2 [ first term + last term]

 

This idea was from the mathematician Carl Friedrich Gauss, who, as a young boy, stunned his teacher by adding up 1 + 2 + 3 + ... + 99 + 100 within a few minutes. Here's how he did it:

He counted 101 terms in the series, of which 50 is the middle term. He also realised that adding the first and last numbers, 1 and 100, gives, 101; and adding the second and second last numbers, 2 and 99, gives 101, as well as 3 + 98 = 101 and so on. Thus he concluded that there are 50 sets of 101 and the middle term is 50. So the sum of the series is:

                   50 (1 + 100) + 50 = 5050.

This can be rewritten as:

                   100/2 (1 + 100) + 50 = 5050   or

                   101/2 (1 + 100) = 5050

 

Arithmetic mean. Given x, y and z are consecutive terms of an A. P., then

                                 y - x = z - y

                                    2y = x + z

                                                      

     y is known as the arithmetic mean.

 

Properties of A. P. (summary of the above points mentioned)

 

Given a sequence u₁, u₂, u₃, ... u_n-1, u_n, u_n+1, ...

 

1. u_n is in the form a + (n - 1)d.

2. u_n - u_n-1 is a constant (common difference).

3. u_n+1 - u_n = u_n - u_n-1

 

 

The next page deals with examples on the application of the abovementioned properties of arithmetic progressions.