Sequence
A succession of numbers arranged in a definite order according to a given certain rule is called sequence. A sequence is either finite or infinite depending upon the number of terms in a sequence.

Series
If a1, a2, a3,…… an is a sequence, then the expression a1 + a2 + a3 + a4 + … + an is called series.

Progression
A sequence whose terms follow certain patterns are more often called progression.

Arithmetic Progression (AP)
A sequence in which the difference of two consecutive terms is constant, is called Arithmetic progression (AP).

  • An arithmetic progression (A.P .) is a sequence in which terms increase or decrease regularly by the same constant. This constant is called  common difference of the A.P. Usually, we denote the first term of A.P . by   a, the common difference by d and the last term by . The general term or the  nth  term of the A.P. is given by 
  • Single Arithmetic mean between any two given numbers a and b: A.M. = 
  •  Arithmetic mean between two given numbers a and b:  form an A.P.
  • If a constant is added to each term of an A.P., then the resulting sequence is also an A.P.
  • If a constant is subtracted to each term of an A.P., then the resulting sequence is also an A.P.
  • If each term of an A.P. is multiplied by a constant, then the resuting sequence is also an A.P.
  • If each term of an A.P. is divided by a constant, then the resuting sequence is also an A.P.
  • Sum of first  terms of an A.P.:  and , where  is the last term, i.e., .

GEOMETRIC PROGRESSION

  • A sequence of non-zero numbers is said to be a geometric progression, if the ratio of each term, except the first one, by its preceding term is always the same. , where  is the first term and  is the common ratio.
  •  term of a G.P.: 
  • Sum of  terms of a G.P.:  if .
  • Sum to infinity of a G.P.: 
  • Geometric mean between a and b: 
  •  Geometric means between a and b: 
  • If all the terms of a G.P. be multiplied or divided by the same quantity the resulting sequence is also a G.P.
  • The reciprocal of the terms of a given G.P. form a G.P.
  • If each term of a G.P. be raised to the same power, the resulting sequence is also a G.P.

ARITHMETIC – GEOMETRIC SERIES

  • A sequence of non-zero numbers is said to be a arithmetic-geometric series, if its terms are obtained on multiplying the terms of an A.P. by the corresponding terms of a G.P. For example:   
  • The general form of an arithmetic-geometric series:
  • nth term of an arithmetic-geometric series:  of A.P. x  of G.P.
  • Sum of n terms of some special series : 
  • Sum of squares of first n natural numbers =
  • Sum of cubes of fist n natural numebrs = 

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