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Arithmetic Progressions - Fundamentals

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An arithmetic progression is a sequence of numbers where the difference between any 2 consecutive terms is a constant. This constant value is called the common difference usually denoted by ‘d’.
If ‘a’ is the first term of the A.P and ‘d’ is the common difference then the terms of the A.P can be represented as a, a + d, a + 2d, a + 3d……
The nth term is usually represented by tn and the sum to n terms is denoted by Sn.

tn = a + (n - 1)d
Sn = (n/2)[2a + (n - 1)d]
= (n/2) [a + (a + (n - 1)d)]
= (n/2) [a + l]
where l denotes the last term of the A.P.

The average of all terms in an A.P is called their arithmetic mean.
Arithmetic Mean (A.M) = Sum of all the terms of the A.P/ Number of terms in the A.P
= Sum of the first and the last terms / 2
= Average of the first and the last terms.

The A.M is also equal to the average of any two numbers which are equidistant from either ends. For example, average of the second term and the penultimate term is also equal to the A.M. Sn can also be calculated from the A.M as follows:
Sn = A.M × n

There are problems which involve three numbers which are in A.P. In such cases, the three numbers can be represented as (a - d), a, and (a + d) so that simplifications will be easier as terms get cancelled out. Similarly, four terms can be represented as (a - 3d), (a - d), (a +d), and (a + 3d) [here the common difference is 2d] and five terms can be represented as (a - 2d), (a - d), a, (a + d), and (a + 2d).


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