CLASS XI Series 6
SEQUENCE
A sequence is an arrangement of numbers in a definite
order according to a certain rule.
Note : A sequence may be finite or infinite according to
their number of terms.
SERIES
Let {tn} be a sequence, then the expression of the form
t 1 + t 2 + .....+ tn + ..... is called a series. The series is finite
or infinite according to the given sequence is finite or
infinite.
ARITHMETIC PROGRESSION (A.P.)
An A.P. is a sequence whose terms either increase or
decrease by a fixed number.
Such a fixed number is called common difference
denoted by ‘d’.
z Let a be the first term, d be the common difference,
l be the last term and n be the number of terms in
an A.P. such that a, a + d, a + 2d, ...... is an A.P.,
then
General term an = a + (n – 1)d or
l = a + (n – 1)d
Sum of first n terms Sn=+n an d−
2
{()} 21
or Sn=+nal
2()We can verify the following simple properties of an A.P.:
z If a constant is added to each term of an A.P., the
resulting sequence is also an A.P.
z If a constant is subtracted from each term of an A.P.,
the resulting sequence is also an A.P.
z If each term of an A.P. is multiplied by a constant,
then the resulting sequence is also an A.P.
z If each term of an A.P. is divided by a non-zero
constant then the resulting sequence is also an A.P.
ARITHMETIC MEAN (A.M.)
z Let a, b be any two numbers and A be the arithmetic
mean between them. Then a, A, b are in A.P.
⇒ A=ab+ 2
z Let a, b be any two numbers and A 1 , A 2 , ....., An be n
A.M.’ s between them. Then a, A 1 , A 2 , ..... , An, b are
in A.P. ⇒ = −
+d ba
n 1
So, Aaba
nAa ba(^121) n
2
1
=+−
- =+ −
⎛
⎝⎜
⎞
, ⎠⎟, ..... ,
Aanba
n n
=+ −
⎛
⎝⎜
⎞
1 ⎠⎟
Sum of n A.M.’s = ⎛ +
⎝⎜
⎞
n ⎠⎟
ab
2
GEOMETRIC PROGRESSION (G.P.)
A G.P. is a succession of numbers in which first term
is non-zero and each next term is the product of its
preceding term and a non-zero constant.
This non-zero constant is called common ratio denoted
by r.
Sequences and Series