CBSE Class 11 Maths Notes Chapter 9 Sequences and Series | Quick revision Notes | EduGrown-Notes

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,…… a_n is a sequence, then the expression a₁ + a₂ + a₃ + a₄ + … + a_n 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  n^th  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:
  
• n^th 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 =