# Progression & Series

Definition: A sequence (progression) is a set of numbers in a definite order with a definite rule of obtaining the numbers.

### Arithmetic Progression (A. P.)

Definition: An A.P. is a sequence whose terms increase or decrease by a fixed number , called the common difference of the A.P.

### nth Term

If a is the first term and d the common difference, the A.P. can be written as a , a + d , a + 2d , ……

The nth term an is given by an = a + (n − 1)d

### Sum of n Terms

The sum Sn of the first n terms of such an A.P. is given by

Sn = ( n/2) [2a + (n − 1)d ]

Sn = ( n/2) [ a + l ]

where l is the last term (i.e. the nth term of the A.P.).

### Notes:

* If a fixed number is added (subtracted) to each term of a given A.P. then the resulting sequence is also an A.P. with the same common difference as that of the given A.P.

* If each term of an A.P. is multiplied by a fixed number (say k) (or divided by a non-zero fixed number), the resulting sequence is also an A.P. with the common difference multiplied by k.

* If a1, a2, a3…..and b1, b2, b3…are two A.P.’s with common differences d and d’ respectively then a1+b1, a2+b2, a3+b3 , …is also an A.P. with common difference d + d’

* If we have to take three terms in an A.P., it is convenient to take them as a − d, a, a + d.
In general , we take a − rd , a − (r − 1)d ,……a − d , a , a + d ,…….a + rd in case we have to take (2r + 1) terms in an A.P

* If we have to take four terms , we take a − 3d , a − d , a + d , a + 3d.
In general, we take a − (2r − 1)d, a − (2r − 3)d,….a − d , a + d,…..a + (2r − 1)d, in case we have to take 2r terms in an A.P.

* If a1, a2, a3, ….. an are in A.P. then a1 + an = a2 + an−1 = a3 + an−2 = . . . . . and so on.

* If nth term of any sequence is a linear expression in n, then the sequence is an AP, whose common difference is the coefficient of n.

* If sum of n terms of any sequence is a quadratic in n, whose constant term is zero, then the sequence is an AP, whose common difference is twice the coefficient of n2. If the constant term is non-zero, then it is an A.P. from second term onwards.

* If {tn} is an A.P., then the common difference, d, is given by d = (tp−tq)/(p−q) , (p , q∈N)

### Arithmetic Mean(s):

* If three terms are in A.P., then the middle term is called the arithmetic mean (A.M.) between the other two i.e. if a, b, c are in A.P. then b = (a + c)/2 is the A.M. of a and c.

* If a1, a2, … an are n numbers then the arithmetic mean (A) of these numbers is * The n numbers A1, A2…..An are said to be A.M.’ s between the numbers a and b if a, A1, A2 ,…..An, b are in A.P. If d is the common difference of this A.P. then b = a + (n + 2 − 1)d  where Ar is the rth mean

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