Arithmetic Progression (or simply A.P.)

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Quantities are said to be in Arithmetic progression when they increase or decrease by a common difference.

Examples:

Each one of the following series form an A.P.

i) 1, 3, 5, 7, …

ii) 3, 7, 11, 15, …

iii) 15, 12, 9, …

iv) x, x - d, x - 2d, .....

The common difference is found by subtracting any term of the series from the immediate succeeding term.

In the above example, common difference in the first is 2, in the second it is 4, in the third it is -3, in the fourth it is -d and in the fifth it is d.

The general form of an A.P. is as follows:

a = first term, d = common difference, then A.P. is a, a+d, a+2d, a+3d,.....

We observe that in any term the coefficient of d is always less by one than the number of terms in the series.

Thus, second term is a+d

third term is a+2d

fourth term is a+3d

tenth term is a+9d

and generally, n^th term is a + (n-1)d.

If n is the number of terms and if t_n is the n^th term, then t_n = a+(n-1)d.

To find the sum of a number of terms in Arithmetical Progression:

Let a=first term, d=common difference, l=t_n=last term, s=required sum. Then,

Writing the series in the reverse order,

Adding together the two series,