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XI-CHAPTER -9:Sequence & Series

LESSON 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 a 1, a 2, a 3,…… an is a sequence, then the expression a 1 + a 2 + a 3 + a 4 + … + 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).

Properties of Arithmetic Progression (AP)
If a sequence is an A.P. then its nth term is a linear expression in n

nth term of an AP ( A n) : If a is the first term, d is common difference and l is the last term of an
AP then
nth term is given by a n = a + (n – 1)d.
nth term of an AP from the last term =a n – (n – 1)d.
Common difference of an AP i.e. d = a n – a n-1 ,∀ n > 1.
If a constant is added or subtracted from each term of an AR then the resulting sequence is an
AP with same common difference.
If each term of an AP is multiplied or divided by a non-zero constant, then the resulting
sequence is also an AP.
If a, b and c are three consecutive terms of an A.P then 2b = a + c.
Any three terms of an AP can be taken as (a – d), a, (a + d) and
any four terms of an AP can be taken as (a – 3d), (a – d), (a + d), (a + 3d)

Sum of n Terms of an AP

Sum of n terms of an AP is given by

S n = [2a + (n – 1)d] = (a 1+ a n) = (a + l ), where l = a n = a + (n – 1)d.
2 2 2
2
A sequence is an AP If the sum of n terms is of the form An + Bn, where A and B are constant
and A = half of common difference i.e. 2A = d.