Differential Equations

SEQUENCE AND SERIES



SEQUENCE AND SERIES
(iii) The common difference can be zero, positive or
1. DEFINITION negative.
Sequence is a function whose domain is the set N of natural (iv) The sum of the two terms of an AP equidistant
numbers. from the beginning & end is constant and equal
to the sum of first & last terms.
Real Sequence : A sequence whose range is a subset of R
is called a real sequence. (v) Any term of an AP (except the first) is equal to
half the sum of terms which are equidistant from
Series : If a1, a2, a3, a4, ........., an, .......... is a sequence, it. an = 1/2 (an – k + an + k), k < n.
then the expression
For k = 1, an = (1/2) (an – 1 + an + 1) ;
a1 + a2 + a3 + a4 + a5 + ........ + ......... + an + ......... is a
series. For k = 2, an = (1/2) (an – 2 + an + 2) and so on.
A series if finite or infinite according as the number of (vi) tr = Sr – Sr – 1
terms in the corresponding sequence is finite or infinite. (vii) If a, b, c are in AP Ÿ 2 b = a + c.
Progressions : It is not necessary that the terms of a (viii) A sequence is an AP, iff its nth terms is of the form
sequence always follow a certain pattern or they are An + B i.e., a linear expression in n. The common
described by some explicit formula for the nth term. Those difference in such a case is A i.e., the coefficient of n.
sequences whose terms follow certain patterns are called
progressions. 1.2 Geometric Progression (GP)
1.1 An Arithmetic Progression (AP)
GP is a sequence of numbers whose first term is non zero
& each of the succeeding terms is equal to the proceeding
AP is a sequence whose terms increase or decrease by a
terms multiplied by a constant. Thus in a GP the ratio of
fixed number. This fixed number is called the common
successive terms is constant. This constant factor is called
difference. If a is the first term & d the common difference,
the COMMON RATIO of the series & obtained by dividing
then AP can be written as a nth term of this AP as
any term by that which immediately proceeds it. Therefore
tn = a + (n – 1) d, where d = an – an – 1.
a, ar, ar2, ar3, ar4, ........... is a GP with a as the first term &
The sum of the first n terms the AP is given by ; r as common ratio.
n n (i) nth term = a rn – 1
Sn 2a  (n  1)d aA .
2 2
a rn  1
where A is the last term. (ii) Sum of the Ist n terms i.e. Sn = ,if r z 1.
r 1
(iii) Sum of an infinite GP when |r| < 1 when n o f
a
rn o 0 if |r| < 1 therefore, Sf = | r | 1 .
1 r
Properties of Arithmetic Progression
(iv) If each term of a GP be multiplied or divided by the
(i) If each term of an A.P. is increased, decreased, same non-zero quantity, the resulting sequence is
multiplied or divided by the same non zero also a GP.
number, then the resulting sequence is also an AP.
(v) Any 3 consecutive terms of a GP can be taken as
(ii) 3 numbers in AP are a – d, a, a + d; a/r, a, ar ; any 4 consecutive terms of a GP can be
4 numbers in AP are a – 3d, a – d, a + d, a + 3d ; taken as a/r3, a/r, ar ar3 & so on.
5 numbers in AP are a – 2d, a – d, a, a + d, a + 2d; (vi) If a, b, c are in GP Ÿ b2 = ac.
6 numbers in AP are a – 5d, a – 3d, a – d, a + d,
a + 3d; a + 5d.